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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
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