Example 8.3.12 (d): Differentiating and Integrating Power Series
Find, with proof, a power series centered at c = 0 for the function f(x) = 1/(1-x)2, for -1 < x < 1.
We need to resort to information we already know. We did discuss the geometric series:
1/1-x = xn
Differentiating both sides finishes the problem:
f(x) = 1/(1-x)2 = 1/1-x = xn = nxn-1
n xn-1 f(x) = 1/(1-x)2