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Example 8.3.12 (e): Differentiating and Integrating Power Series

Find a simple expression for n xn and n2 xn, where |x| < 1. How about n3 xn

Let's look at the first series term-by-term:

n xn = x + 2 x2 + 3 x3 + 4 x4 + ... =
     = x (1 + 2 x + 3 x2 + 4 x3 + ...) =
     = x (x + x2 + x3 + x3 + ...) =
     = x 1/1-x =
     = x/(1-x)2

As usual, this is confirmed by looking at the plots.

n xn f(x) = x/(1-x)2

The answer to the second question is:

n2 xn = x(x+1)/(1-x)3

but why, oh why?

n2 xn f(x) = x(x+1)/(1-x)3

For the last question you're on your own entirely ...

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