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For this purpose take Maclaurin’s Theorem,
$$\begin{array}{cc}f(u)=f(0)+uf\prime (0)+\frac{1}{2!}{u}^{2}f\prime \prime (0)+\frac{1}{3!}{u}^{3}f\prime \prime \prime (0)+\dots ,& \\ & \end{array}$$and put
$$\begin{array}{cc}f(u)=sinhu,\phantom{\rule{1em}{0ex}}f\prime (u)=coshu,\phantom{\rule{1em}{0ex}}f\prime \prime (u)=sinhu,\dots ,& \\ & \end{array}$$then
$$\begin{array}{cc}f(0)=sinh0=0,\phantom{\rule{1em}{0ex}}f\prime (0)=cosh0=1,\dots ;& \\ & \end{array}$$hence
$$\begin{array}{cc}sinhu=u+\frac{1}{3!}{u}^{3}+\frac{1}{5!}{u}^{5}+\dots ;& \text{(27)}\end{array}$$and similarly, or by diﬀerentiation,
$$\begin{array}{cc}coshu=1+\frac{1}{2!}{u}^{2}+\frac{1}{4!}{u}^{4}+\dots .& \text{(28)}\end{array}$$By means of these series the numerical values of $sinhu,coshu$, can be computed and tabulated for successive values of the independent variable $u$. They are convergent for all values of $u$, because the ratio of the $n$th term to the preceding is in the ﬁrst case $\frac{{u}^{2}}{(2n-1)(2n-2)}$, and in the second case $\frac{{u}^{2}}{(2n-2)(2n-3)}$, both of which ratios can be made less than unity by taking $n$ large enough, no matter what value $u$ has. Lagrange’s remainder shows equivalence of function and series.
From these series the following can be obtained by division:
These four developments are seldom used, as there is no observable law in the coeﬃcients, and as the functions $tanhu,sechu,cothu,cschu$, can be found directly from the previously computed values of $coshu,sinhu$.
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