## Example 8.3.12 (e): Differentiating and Integrating Power Series

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

n x^{n}= x + 2 x^{2}+ 3 x^{3}+ 4 x^{4}+ ... =

= x (1 + 2 x + 3 x^{2}+ 4 x^{3}+ ...) =

= x (x + x^{2}+ x^{3}+ x^{3}+ ...) =

= x^{1}/_{1-x}=

=^{x}/_{(1-x)2}

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

n x^{n}f(x) =^{x}/_{(1-x)2}

The answer to the second question is:

n^{2}x^{n}=^{x(x+1)}/_{(1-x)3}

but why, oh why?

n^{2}x^{n}f(x) =^{x(x+1)}/_{(1-x)3}

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