Example 8.4.18 (c): Finding Taylor Series by Differentiation
ExampleFind a series for f(x) = 2x/(1-x2)2
This time we need to find a function whose derivative is our function in question, and whose Taylor series we know. Experimenting on scratch paper for a little while will give us, as a guess, the function
g(x) = 1/1-x2
The derivative of this function is g'(x) = 2x/(1-x2)2, our function in question. Thus:
f(x) = g'(x) = g(x) = x2n = x2n = 2n x2n-1
for |x| < 1. We have, of course, quietly used substitution for the Geometric series and as a result - voila! - we have a Taylor series centered at zero for our function, just as desired.
2n x2n-1 f(x) = 2x/(1-x2)2