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196. THE number of essentially different types of functions with which we have been concerned in the foregoing chapters is not very large. Among those which have occurred the most important for ordinary purposes are polynomials, rational functions, algebraical functions, explicit or implicit, and trigonometrical functions, direct or inverse.
We are however far from having exhausted the list of functions which are important in mathematics. The gradual expansion of the range of mathematical knowledge has been accompanied by the introduction into analysis of one new class of function after another. These new functions have generally been introduced because it appeared that some problem which was occupying the attention of mathematicians was incapable of solution by means of the functions already known. The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised. One of the most fruitful sources of new functions has been the problem of integration. Attempts have been made to integrate some function in terms of functions already known. These attempts have failed; and after a certain number of failures it has begun to appear probable that the problem is insoluble. Sometimes it has been proved that this is so; but as a rule such a strict proof has not been forthcoming until later on. Generally it has happened that mathematicians have taken the impossibility for granted as soon as they have become reasonably convinced of it, and have introduced a new function defined by its possessing the required property, viz. that . Starting from this definition, they have investigated the properties of ; and it has then appeared that has properties which no finite combination of the functions previously known could possibly have; and thus the correctness of the assumption that the original problem could not possibly be solved has been established. One such case occurred in the preceding pages, when in Ch. VI we defined the function by means of the equation
Let us consider what grounds we have for supposing to be a really new function. We have seen already (Ex. XLII. 4) that it cannot be a rational function, since the derivative of a rational function is a rational function whose denominator contains only repeated factors. The question whether it can be an algebraical or trigonometrical function is more difficult. But it is very easy to become convinced by a few experiments that differentiation will never get rid of algebraical irrationalities. For example, the result of differentiating any number of times is always the product of by a rational function, and so generally. The reader should test the correctness of the statement by experimenting with a number of examples. Similarly, if we differentiate a function which involves or , one or other of these functions persists in the result.
We have, therefore, not indeed a strict proof that is a new function—that we do not profess to give97—but a reasonable presumption that it is. We shall therefore treat it as such, and we shall find on examination that its properties are quite unlike those of any function which we have as yet encountered.
197. Definition of . We define , the logarithm of , by the equation
We must suppose that is positive, since (Ex. LXXVI. 2) the integral has no meaning if the range of integration includes the point . We might have chosen a lower limit other than ; but proves to be the most convenient. With this definition .We shall now consider how behaves as varies from towards . It follows at once from the definition that is a continuous function of which increases steadily with and has a derivative
and it follows from § 175 that tends to as .If is positive but less than , then is negative. For
Moreover, if we make the substitution in the integral, we obtain Thus tends steadily to as decreases from to .The general form of the graph of the logarithmic function is shown in Fig. 52. Since the derivative of is , the slope of
the curve is very gentle when is very large, and very steep when is very small.
Examples LXXXII. 1. Prove from the definition that if then
[For , and the subject of integration lies between and .]
2. Prove that lies between and when is positive. [Use the fact that .]
3. If then .
4. Prove that
[Use Ex. 1.]
198. The functional equation satisfied by . The function satisfies the functional equation
(1) |
For, making the substitution , we see that
which proves the theorem.
Examples LXXXIII. 1. It can be shown that there is no solution of the equation (1) which possesses a differential coefficient and is fundamentally distinct from . For when we differentiate the functional equation, first with respect to and then with respect to , we obtain the two equations
and so, eliminating , . But if this is true for every pair of values of and , then we must have , or , where is a constant. Hence and it is easy to see that . Thus there is no solution fundamentally distinct from , except the trivial solution , obtained by taking .2. Show in the same way that there is no solution of the equation
which possesses a differential coefficient and is fundamentally distinct from .199. The manner in which tends to infinity with . It will be remembered that in Ex. XXXVI. 6 we defined certain different ways in which a function of may tend to infinity with , distinguishing between functions which, when is large, are of the first, second, third, … orders of greatness. A function was said to be of the th order of greatness when tends to a limit different from zero as tends to infinity.
It is easy to define a whole series of functions which tend to infinity with , but whose order of greatness is smaller than the first. Thus , , , … are such functions. We may say generally that , where is any positive rational number, is of the th order of greatness when is large. We may suppose as small as we please, e.g. less than . And it might be thought that by giving all possible values we should exhaust the possible ‘orders of infinity’ of . At any rate it might be supposed that if tends to infinity with , however slowly, we could always find a value of so small that would tend to infinity more slowly still; and, conversely, that if tends to infinity with , however rapidly, we could always find a value of so great that would tend to infinity more rapidly still.
Perhaps the most interesting feature of the function is its behaviour as tends to infinity. It shows that the presupposition stated above, which seems so natural, is unfounded. The logarithm of tends to infinity with , but more slowly than any positive power of , integral or fractional. In other words but
for all positive values of . This fact is sometimes expressed loosely by saying that the ‘order of infinity of is infinitely small’; but the reader will hardly require at this stage to be warned against such modes of expression.200. Proof that as . Let be any positive number. Then when , and so
or when . Now if is any positive number we can choose a smaller positive value of . And then But, since , as , and therefore201. The behaviour of as . Since
if , it follows from the theorem proved above that Thus tends to and to as tends to zero by positive values, but tends to more slowly than any positive power of , integral or fractional.202. Scales of infinity. The logarithmic scale. Let us consider once more the series of functions
which possesses the property that, if and are any two of the functions contained in it, then and both tend to as , while tends to or to according as occurs to the right or the left of in the series. We can now continue this series by the insertion of new terms to the right of all those already written down. We can begin with , which tends to infinity more slowly than any of the old terms. Then tends to more slowly than , than , and so on. Thus we obtain a series formed of two simply infinite series arranged one after the other. But this is not all. Consider the function , the logarithm of . Since , for all positive values of , it follows on putting that Thus tends to with , but more slowly than any power of . Hence we may continue our series in the formand it will by now be obvious that by introducing the functions , , … we can prolong the series to any extent we like. By putting we obtain a similar scale of infinity for functions of which tend to as tends to by positive values.98
Examples LXXXIV. 1. Between any two terms , of the series we can insert a new term such that tends to more slowly than and more rapidly than . [Thus between and we could insert : between and we could insert . And, generally, satisfies the conditions stated.]
2. Find a function which tends to more slowly than , but more rapidly than , where is any rational number less than . [ is such a function; or , where is any positive rational number.]
3. Find a function which tends to more slowly than , but more rapidly than , where is any rational number. [The function is such a function. It will be gathered from these examples that incompleteness is an inherent characteristic of the logarithmic scale of infinity.]
4. How does the function
behave as tends to ? [If then the behaviour of is dominated by that of . If then the power of disappears and the behaviour of is dominated by that of , unless , when it is dominated by that of . Thus if , or , , or , , , and if , or , , or , , .]5. Arrange the functions , , , according to the rapidity with which they tend to infinity as .
6. Arrange
according to the rapidity with which they tend to zero as .7. Arrange
according to the rapidity with which they tend to zero as .8. Show that
and so on.9. Show that
and so on.203. The number . We shall now introduce a number, usually denoted by , which is of immense importance in higher mathematics. It is, like , one of the fundamental constants of analysis.
We define as the number whose logarithm is . In other words is defined by the equation
Since is an increasing function of , in the stricter sense of § 95, it can only pass once through the value . Hence our definition does in fact define one definite number.Now and so
where is any positive integer. Hence Again, if and are any positive integers, and denotes the positive th root of , we have so that . Thus, if has any positive rational value, and denotes the positive th power of , we have(1) |
and . Hence the equation (1) is true for all rational values of , positive or negative. In other words the equations
(2) |
are consequences of one another so long as is rational and has its positive value. At present we have not given any definition of a power such as in which the index is irrational, and the function is defined for rational values of only.
Example. Prove that . [In the first place it is evident that
and so . Also so that .]204. The exponential function. We now define the exponential function for all real values of as the inverse of the logarithmic function. In other words we write
if .We saw that, as varies from towards , increases steadily, in the stricter sense, from towards . Thus to one value of corresponds one value of , and conversely. Also is a continuous function of , and it follows from § 109 that is likewise a continuous function of .
It is easy to give a direct proof of the continuity of the exponential function. For if and then
Thus is greater than if , and than if ; and if is very small must also be very small.Thus is a positive and continuous function of which increases steadily from towards as increases from towards . Moreover is the positive th power of the number , in accordance with the elementary definitions, whenever is a rational number. In particular when . The general form of the graph of is as shown in Fig. 53.
205. The principal properties of the exponential function. (1) If , so that , then and
Thus the derivative of the exponential function is equal to the function itself . More generally, if then .(2) The exponential function satisfies the functional equation
This follows, when and are rational, from the ordinary rules of indices. If or , or both, are irrational then we can choose two sequences , , …, , … and , , …, , … of rational numbers such that , . Then, since the exponential function is continuous, we have
In particular , or .We may also deduce the functional equation satisfied by from that satisfied by . For if , , so that , , then and
Examples LXXXV. 1. If then , where is a constant.
2. There is no solution of the equation fundamentally distinct from the exponential function. [We assume that has a differential coefficient. Differentiating the equation with respect to and in turn, we obtain
and so , and therefore each is constant. Thus if then , where is a constant, so that (Ex. 1).]3. Prove that as . [Applying the Mean Value Theorem, we obtain , where .]
206. (3) The function tends to infinity with more rapidly than any power of , or as , for all values of however great.
We saw that as , for any positive value of however small. Writing for , we see that for any value of however large. The result follows on putting . It is clear also that tends to if , and to if , and in each case more rapidly than any power of .
From this result it follows that we can construct a ‘scale of infinity’ similar to that constructed in §202, but extending in the opposite direction; i.e. a scale of functions which tend to more and more rapidly as .99 The scale is
where of course , …, , … denote , …, , ….The reader should try to apply the remarks about the logarithmic scale, made in §202 and Exs. LXXXIV, to this ‘exponential scale’ also. The two scales may of course (if the order of one is reversed) be combined into one scale
207. The general power . The function has been defined only for rational values of , except in the particular case when . We shall now consider the case in which is any positive number. Suppose that is a positive rational number . Then the positive value of the power is given by ; from which it follows that
and so We take this as our definition of when is irrational. Thus . It is to be observed that , when is irrational, is defined only for positive values of , and is itself essentially positive; and that . The most important properties of the function are as follows.(1) Whatever value may have, and . In other words the laws of indices hold for irrational no less than for rational indices. For, in the first place,
and in the second(2) If then , where is positive. The graph of is in this case similar to that of , and as , more rapidly than any power of .
If then , where is positive. The graph of is then similar in shape to that of , but reversed as regards right and left, and as , more rapidly than any power of .
(3) is a continuous function of , and
(4) is also a continuous function of , and
(5) as . This of course is a mere corollary from the fact that , but the particular form of the result is often useful; it is of course equivalent to the result (Ex. LXXXV. 3) that as .
In the course of the preceding chapters a great many results involving the function have been stated with the limitation that is rational. The definition and theorems given in this section enable us to remove this restriction.
208. The representation of as a limit. In Ch. IV, § 73, we proved that tends, as , to a limit which we denoted provisionally by . We shall now identify this limit with the number of the preceding sections. We can however establish a more general result, viz. that expressed by the equations
(1) |
As the result is of very great importance, we shall indicate alternative lines of proof.
(1) Since
it follows that If we put , we see that as or . Since the exponential function is continuous it follows that as or : i.e. that(2) |
If we suppose that or through integral values only, we obtain the result expressed by the equations (1).
(2) If is any positive integer, however large, and , we have
or(3) |
Writing for , so that is positive and , we obtain, after some simple transformations,
(4) |
Now let
Then , at any rate for sufficiently large values of ; and, by (9) of §74, which evidently tends to as . The result now follows from the inequalities (4). The more general result (2) may be proved in the same way, if we replace by a continuous variable .209. The representation of as a limit. We can also prove (cf. §75) that
For
which tends to zero as , since tends to a limit (§75) and to (Ex. XXVII. 10). The result now follows from the inequalities (3) of §208.Examples LXXXVI. 1. Prove, by taking and in the inequalities (4) of §208, that .
2. Prove that if then , and so that if then
Hence deduce the results of §209.3. If is a function of such that as , then . [Writing in the form
and using Ex. LXXXII. 4, we see that .]4. If , then ; and if and , then
5. Deduce from (1) of §208 the theorem that tends to infinity more rapidly than any power of .
210. Common logarithms. The reader is probably familiar with the idea of a logarithm and its use in numerical calculation. He will remember that in elementary algebra , the logarithm of to the base , is defined by the equations
This definition is of course applicable only when is rational, though this point is often passed over in silence.Our logarithms are therefore logarithms to the base . For numerical work logarithms to the base are used. If
then and also , so that Thus it is easy to pass from one system to the other when once has been calculated.It is no part of our purpose in this book to go into details concerning the practical uses of logarithms. If the reader is not familiar with them he should consult some text-book on Elementary Algebra or Trigonometry.100
Examples LXXXVII. 1. Show that
where , , . Hence determine the th derivatives of the functions , , and show in particular that .2. Trace the curve , where and are positive. Show that has an infinity of maxima whose values form a geometrical progression and which lie on the curve
3. Integrals containing the exponential function. Prove that
[Denoting the two integrals by , , and integrating by parts, we obtain
Solve these equations for and .]4. Prove that the successive areas bounded by the curve of Ex. 2 and the positive half of the axis of form a geometrical progression, and that their sum is
5. Prove that if then
6. If then . [Integrate by parts. It follows that can be calculated for all positive integral values of .]
7. Prove that, if is a positive integer, then
and8. Show how to find the integral of any rational function of . [Put , when , , and the integral is transformed into that of a rational function of .]
9. Integrate
distinguishing the cases in which is and is not equal to .10. Prove that we can integrate any function of the form , where denotes a polynomial. [This follows from the fact that can be expressed as the sum of a number of terms of the type , where is a positive integer.]
11. Show how to integrate any function of the form
12. Prove that , where and is greater than the greatest root of the denominator of , is convergent. [This follows from the fact that tends to infinity more rapidly than any power of .]
13. Prove that , where , is convergent for all values of , and that the same is true of , where is any positive integer.
14. Draw the graphs of , , , , , , and , determining any maxima and minima of the functions and any points of inflexion on their graphs.
15. Show that the equation , where and are positive, has two real roots, one, or none, according as , , or . [The tangent to the curve at the point is
which passes through the origin if , so that the line touches the curve at the point . The result now becomes obvious when we draw the line . The reader should discuss the cases in which or or both are negative.]16. Show that the equation has no real root except , and that has three real roots.
17. Draw the graphs of the functions
18. Determine roughly the positions of the real roots of the equations
19. The hyperbolic functions. The hyperbolic functions ,101 , … are defined by the equations
Draw the graphs of these functions.
20. Establish the formulae
21. Verify that these formulae may be deduced from the corresponding formulae in and , by writing for and for .
[It follows that the same is true of all the formulae involving and which are deduced from the corresponding elementary properties of and . The reason of this analogy will appear in Ch. X.]
22. Express and in terms (a) of (b) of . Discuss any ambiguities of sign that may occur.
23. Prove that
[All these formulae may of course be transformed into formulae in integration.]
24. Prove that and .
25. Prove that if then , if then , and if then . Account for the ambiguity of sign in the first case.
26. We shall denote the functions inverse to , , by , , . Show that is defined only when , and is in general two-valued, while is defined for all real values of , and when , and both of the two latter functions are one-valued. Sketch the graphs of the functions.
27. Show that if and is positive, and , then
28. Prove that if then , and is equal to or to , according as or .
29. Prove that if then is equal to or to , according as is less than or greater than . [The results of Exs. 28 and 29 furnish us with an alternative method of writing a good many of the formulae of Ch. VI.]
30. Prove that
31. Prove that
32. Solve the equation , where , showing that it has no real roots if , while if it has two, one, or no real roots according as and are both positive, of opposite signs, or both negative. Discuss the case in which .
33. Solve the simultaneous equations , .
34. as . [For , and . Cf. Ex. XXVII. 11.] Show also that the function has a maximum when , and draw the graph of the function for positive values of .
36. If , where , as , then . [For , and so (Ch. IV, Misc. Ex. 27).]
37. as .
[If then . Now use Ex. 36.]
38. as .
39. Discuss the approximate solution of the equation .
[It is easy to see by general graphical considerations that the equation has two positive roots, one a little greater than and one very large,102 and one negative root a little greater than . To determine roughly the size of the large positive root we may proceed as follows. If then
roughly, since and are approximate values of and respectively. It is easy to see from these equations that the ratios and do not differ greatly from unity, and that gives a tolerable approximation to the root, the error involved being roughly measured by or or , which is less than . The approximations are of course very rough, but suffice to give us a good idea of the scale of magnitude of the root.]40. Discuss similarly the equations
211. Logarithmic tests of convergence for series and integrals. We showed in Ch. VIII (§§ 175 et seq.) that
are convergent if and divergent if . Thus is divergent, but is convergent for all positive values of .We saw however in § 200 that with the aid of logarithms we can construct functions which tend to zero, as , more rapidly than , yet less rapidly than , however small may be, provided of course that it is positive. For example is such a function, and the question as to whether the series
is convergent or divergent cannot be settled by comparison with any series of the type .The same is true of such series as
It is a question of some interest to find tests which shall enable us to decide whether series such as these are convergent or divergent; and such tests are easily deduced from the Integral Test of § 174.For since
we have if . The first integral tends to the limit as , if , and to if . The second integral tends to . Hence the series and integral where and are greater than unity, are convergent if , divergent if .It follows, of course, that is convergent if is positive and less than , where , for all values of greater than some definite value, and divergent if is positive and greater than for all values of greater than some definite value. And there is a corresponding theorem for integrals which we may leave to the reader.
Examples LXXXVIII. 1. The series
are convergent. [The convergence of the first series is a direct consequence of the theorem of the preceding section. That of the second follows from the fact that is less than for sufficiently large values of , however small may be, provided that it is positive. And so, taking , is less than for sufficiently large values of . The convergence of the third series follows from the comparison test at the end of the last section.]2. The series
are divergent.3. The series
where , are convergent for all values of and ; similarly the series are divergent.4. The question of the convergence or divergence of such series as
cannot be settled by the theorem of p. 1267, since in each case the function under the sign of summation tends to zero more rapidly than yet less rapidly than , where is any positive number however small. For such series we need a still more delicate test. The reader should be able, starting from the equationswhere , , …, to prove the following theorem: the series and integral are convergent if and divergent if , and being any numbers sufficiently great to ensure that and are positive when or . These values of and increase very rapidly as increases: thus requires , requires , requires , and so on; and it is easy to see that , , .
The reader should observe the extreme rapidity with which the higher exponential functions, such as and , increase with . The same remark of course applies to such functions as and , where has any value greater than unity. It has been computed that has figures, while has of course . Conversely, the rate of increase of the higher logarithmic functions is extremely slow. Thus to make we have to suppose a number with over figures.103
5. Prove that the integral , where , is convergent if , divergent if . [Consider the behaviour of
as . This result also may be refined upon by the introduction of higher logarithmic factors.]6. Prove that has no meaning for any value of . [The last example shows that is a necessary condition for convergence at the lower limit: but tends to like , as , if is negative, and so the integral diverges at the upper limit when .]
7. The necessary and sufficient conditions for the convergence of are , .
Examples LXXXIX. 1. Euler’s limit. Show that
tends to a limit as , and that . [This follows at once from §174. The value of is in fact , and is usually called Euler’s constant.]2. If and are positive then
tends to a limit as .3. If then
tends to a limit as .4. Show that the series
is divergent. [Compare the general term of the series with .] Show also that the series derived from , in the same way that the above series is derived from , is convergent if and otherwise divergent.5. Prove generally that if is a series of positive terms, and
then is convergent or divergent according as is convergent or divergent. [If is convergent then tends to a positive limit , and so is convergent. If is divergent then , and (Ex. LXXXII. 1); and it is evident that tends to as .]6. Prove that the same result holds for the series . [The proof is the same in the case of convergence. If is divergent, and from a certain value of onwards, then , and the divergence of follows from that of . If on the other hand for an infinity of values of , as might happen with a rapidly divergent series, then for all these values of .]
7. Sum the series . [We have
by Ex. 1, denoting Euler’s constant, and , being numbers which tend to zero as . Subtracting and making we see that the sum of the given series is . See also §213.]8. Prove that the series
oscillates finitely except when , when it converges.212. Series connected with the exponential and logarithmic functions. Expansion of by Taylor’s Theorem. Since all the derivatives of the exponential function are equal to the function itself, we have
where . But as , whatever be the value of (Ex. XXVII. 12); and . Hence, making tend to , we have(1) |
The series on the right-hand side of this equation is known as the exponential series. In particular we have
(2) |
and so
(3) |
a result known as the exponential theorem. Also
(4) |
for all positive values of .
The reader will observe that the exponential series has the property of reproducing itself when every term is differentiated, and that no other series of powers of would possess this property: for some further remarks in this connection see Appendix II.
The power series for is so important that it is worth while to investigate it by an alternative method which does not depend upon Taylor’s Theorem. Let
and suppose that . Then which is less than . And, provided , we have also, by the binomial theorem for a negative integral exponent, Thus But (§208) the first and last functions tend to the limit as , and therefore must do the same. From this the equation (1) follows when is positive; its truth when is negative follows from the fact that the exponential series, as was shown in Ex. LXXXI. 7, satisfies the functional equation , so that .2. If is positive then the greatest term in the exponential series is the -th, unless is an integer, when the preceding term is equal to it.
3. Show that . [For is one term in the series for .]
4. Prove that , where
and ; and deduce that lies between and .5. Employ the exponential series to prove that tends to infinity more rapidly than any power of . [Use the inequality .]
6. Show that is not a rational number. [If , where and are integers, we must have
or, multiplying up by , and this is absurd, since the left-hand side is integral, and the right-hand side less than .]7. Sum the series , where is a polynomial of degree in . [We can express in the form
and8. Show that
and that if then In particular the last series is equal to zero when .9. Prove that , , , and that , where is any positive integer, is a positive integral multiple of .
10. Prove that .
[Multiply numerator and denominator by , and proceed as in Ex. 7.]
11. Determine , , so that tends to a limit as , evaluate the limit, and draw the graph of the function .
12. Draw the graphs of , , , and compare them with that of .
13. Prove that is positive or negative according as is odd or even. Deduce the exponential theorem.
14. If
then . Hence prove that if then and generally . Deduce the exponential theorem.15. Show that the expansion in powers of of the positive root of begins with the terms
213. The logarithmic series. Another very important expansion in powers of is that for . Since
and if is numerically less than unity, it is natural to expect104 that will be equal, when , to the series obtained by integrating each term of the series from to , i.e. to the series . And this is in fact the case. For and so, if , whereWe require to show that the limit of , when tends to , is zero. This is almost obvious when ; for then is positive and less than
and therefore less than . If on the other hand , we put and , so that which shows that has the sign of . Also, since the greatest value of in the range of integration is , we have and so .Hence
provided that . If lies outside these limits the series is not convergent. If we obtain a result already proved otherwise (Ex. LXXXIX. 7).214. The series for the inverse tangent. It is easy to prove in a similar manner that
provided that . The only difference is that the proof is a little simpler; for, since is an odd function of , we need only consider positive values of . And the series is convergent when as well as when . We leave the discussion to the reader. The value of which is represented by the series is of course that which lies between and when , and which we saw in Ch. VII (Ex. LXIII. 3) to be the value represented by the integral. If , we obtain the formula
2. if .
3. Prove that if is positive then
4. Obtain the series for and by means of Taylor’s theorem.
[A difficulty presents itself in the discussion of the remainder in the first series when is negative, if Lagrange’s form is used; Cauchy’s form, viz.
should be used (cf. the corresponding discussion for the Binomial Series, Ex. LVI. 2 and §163).In the case of the second series we have
(Ex. XLV. 11), and there is no difficulty about the remainder, which is obviously not greater in absolute value than .105]
5. If then
[Use the identity . This series may be used to calculate , a purpose for which the series , owing to the slowness of its convergence, is practically useless. Put and find to places of decimals.]
6. Find to places of decimals from the formula
7. Prove that
if , and that if . Given that and , show, by putting in the second formula, that .8. Show that if , , and are known, then the formula
gives with an error practically equal to .9. Show that
where , , .[These formulae enable us to find , , and rapidly and with any degree of accuracy.]
10. Show that
and calculate to places of decimals.11. Show that the expansion of in powers of begins with the terms .
12. Show that
approximately, for large values of . Apply the formula, when , to obtain an approximate value of , and estimate the accuracy of the result.13. Show that
if . [Use Ex. LXXXI. 2.]14. Using the logarithmic series and the facts that and , show that an approximate solution of the equation is .
15. Expand and in powers of as far as , and verify that, to this order,
16. Show that
if . Deduce that[Proceed as in §214 and use the result of Ex. XLVIII. 7.]
17. Prove similarly that
18. Prove generally that if and are positive integers then
and so that the sum of the series can be found. Calculate in this way the sums of and .215. The Binomial Series. We have already (§ 163) investigated the Binomial Theorem
assuming that and that is rational. When is irrational we haveso that the rule for the differentiation of remains the same, and the proof of the theorem given in § 163 retains its validity. We shall not discuss the question of the convergence of the series when or .106
Examples XCII. 1. Prove that if then
2. Approximation to quadratic and other surds. Let be a quadratic surd whose numerical value is required. Let be the square nearest to ; and let or , being positive. Since cannot be greater than , is comparatively small and the surd can be expressed in a series
which is at any rate fairly rapidly convergent, and may be very rapidly so. ThusLet us consider the error committed in taking (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than , which is less than .
3. If is small compared with then
the error being of the order . Apply the process to .[Expanding by the binomial theorem, we have
the error being less than the numerical value of the next term, viz. . Also the error being less than . The result follows. The same method may be applied to surds other than quadratic surds, e.g. to .]4. If differs from by less than per cent. of either then differs from by less than .
5. If , and is small compared with , then a good approximation for is
Show that when , , this approximation is accurate to places of decimals.6. Show how to sum the series
where is a polynomial of degree in .[Express in the form as in Ex. XC. 7.]
7. Sum the series , and prove that
216. An alternative method of development of the theory of the exponential and logarithmic functions. We shall now give an outline of a method of investigation of the properties of and entirely different in logical order from that followed in the preceding pages. This method starts from the exponential series . We know that this series is convergent for all values of , and we may therefore define the function by the equation
(1) |
We then prove, as in Ex. LXXXI. 7, that
(2) |
Again
where is numerically less than so that as . And so as , or(3) |
Incidentally we have proved that is a continuous function.
We have now a choice of procedure. Writing and observing that , we have
and, if we define the logarithmic function as the function inverse to the exponential function, we are brought back to the point of view adopted earlier in this chapter.But we may proceed differently. From (2) it follows that if is a positive integer then
If is a positive rational fraction , then and so is equal to the positive value of . This result may be extended to negative rational values of by means of the equation and so we have say, where for all rational values of . Finally we define , when is irrational, as being equal to . The logarithm is then defined as the function inverse to or .Example. Develop the theory of the binomial series
where , in a similar manner, starting from the equation (Ex. LXXXI. 6).
1. Given that and that and are nearly equal to powers of , calculate and to four places of decimals.
2. Determine which of and is the greater. [Take logarithms and observe that .]
3. Show that cannot be a rational number if is any positive integer not a power of . [If is not divisible by , and , we have , which is impossible, since ends with and does not. If , where is not divisible by , then and therefore
cannot be rational.]4. For what values of are the functions , , , … (a) equal to (b) equal to (c) not defined? Consider also the same question for the functions , , , …, where .
5. Show that
is negative and increases steadily towards as increases from towards .[The derivative of the function is
as is easily seen by splitting up the right-hand side into partial fractions. This expression is positive, and the function itself tends to zero as , since where , and .]6. Prove that
7. If then .
[Put , and use the fact that when .]
8. Show that and both decrease steadily as increases from towards .
9. Show that, as increases from towards , the function assumes once and only once every value between and .
10. Show that as .
11. Show that decreases steadily from to as increases from towards . [The function is undefined when , but if we attribute to it the value when it becomes continuous for . Use Ex. 7 to show that the derivative is negative.]
12. Show that the function , where is positive, decreases steadily as increases from to , and find its limit as .
13. Show that , where and are large positive numbers, if is greater than the greater of and .
[It is easy to prove that ; and so the inequality given is certainly satisfied if
and therefore certainly satisfied if , .]14. If and tend to infinity as , and , then . [Use the result of Ch. VI, Misc. Ex. 33.] By taking , , prove that for all positive values of .
15. If and are positive integers then
as . [Cf. Ex. LXXVIII. 6.]16. Prove that if is positive then as . [We have
where . Now use §209 and Ex. LXXXII. 4.]17. Prove that if and are positive then
[Take logarithms and use Ex. 16.]
18. Show that
where is Euler’s constant (Ex. LXXXIX. 1) and as .19. Show that
the series being formed from the series by taking alternately two positive terms and then one negative. [The sum of the first terms iswhere and tend to as . (Cf. Ex. LXXVIII. 6).]
20. Show that .
21. Prove that
where , . Hence prove that the sum of the series when continued to infinity is22. Show that
23. Prove that the sums of the four series
are , , , respectively.24. Prove that tends to or to according as or .
[If then . It can be shown that the function tends to when : for a proof, which is rather beyond the scope of the theorems of this chapter, see Bromwich’s Infinite Series, pp. 461 et seq.]
25. Find the limit as of
distinguishing the different cases which may arise.26. Prove that
diverges to . [Compare with .] Deduce that if is positive then as . [The logarithm of the function is .]27. Prove that if then
[The difference between and the sum of the first terms of the series is
28. No equation of the type
where , , … are polynomials and , , … different real numbers, can hold for all values of . [If is the algebraically greatest of , , …, then the term outweighs all the rest as .]29. Show that the sequence
tends to infinity more rapidly than any member of the exponential scale.[Let , , and so on. Then, if is any member of the exponential scale, when .]
30. Prove that
where is to be put equal to and to after differentiation. Establish a similar rule for the differentiation of .31. Prove that if then (i) is a polynomial of degree , (ii) , and (iii) all the roots of are real and distinct, and separated by those of . [To prove (iii) assume the truth of the result for , , …, , and consider the signs of for the values of for which and for large (positive or negative) values of .]
32. The general solution of , where is a differentiable function, is , where is a constant: and that of
is or , according as is positive or negative. [In proving the second result assume that has derivatives of the first three orders. Then where and tend to zero with . It follows that , , , so that or .]33. How do the functions , , behave as ?
34. Trace the curves , .
35. The equation has one real root if or , . If then it has two real roots or none, according as or .
36. Show by graphical considerations that the equation has one, two, or three real roots if , none, one, or two if ; and show how to distinguish between the different cases.
37. Trace the curve , showing that the point is a centre of symmetry, and that as increases through all real values, steadily increases from to . Deduce that the equation
has no real root unless , and then one, whose sign is the same as that of . [In the first place is clearly an odd function of . Also The function inside the large bracket tends to zero as ; and its derivative is which has the sign of . Hence for all values of .]38. Trace the curve , and show that the equation
has no real roots if is negative, one negative root if and two positive roots and one negative if .39. Show that the equation has one real root if is odd and none if is even.
[Assume this proved for , , … . Then has at least one real root, since its degree is odd, and it cannot have more since, if it had, or would have to vanish once at least. Hence has just one root, and so cannot have more than two. If it has two, say and , then or must vanish once at least between and , say at . And
But is also positive when is large (positively or negatively), and a glance at a figure will show that these results are contradictory. Hence has no real roots.]40. Prove that if and are positive and nearly equal then
approximately, the error being about . [Use the logarithmic series. This formula is interesting historically as having been employed by Napier for the numerical calculation of logarithms.]41. Prove by multiplication of series that if then
42. Prove that
where with .43. The first terms in the expansion of in powers of are
44. Show that the expansion of
in powers of begins with the terms45. Show that if then
[Use the method of Ex. XCII. 6. The results are more easily obtained by differentiation; but the problem of the differentiation of an infinite series is beyond our range.]
46. Prove that
provided that and are positive. Deduce, and verify independently, that each of the functions
is positive for all positive values of .47. Prove that if , , are all positive, and , then
while if is positive and the value of the integral is that value of the inverse tangent being chosen which lies between and . Are there any other really different cases in which the integral is convergent?48. Prove that if then
and deduce that the value of the integral is if , and if . Discuss the case in which .49. Transform the integral , where , in the same ways, showing that its value is
50. Prove that
51. If , , then
52. Prove that if then
53. Prove that
and deduce that if then[Use the substitutions and .]
54. Prove that
if . [Integrate by parts.]up | next | prev | ptail | top |