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 =
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 ...