For this purpose take Maclaurin’s Theorem,
and similarly, or by diﬀerentiation,
By means of these series the numerical values of , can be computed and tabulated for successive values of the independent variable . They are convergent for all values of , because the ratio of the th term to the preceding is in the ﬁrst case , and in the second case , both of which ratios can be made less than unity by taking large enough, no matter what value 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 , can be found directly from the previously computed values of .