1. ## Power Series

$\displaystyle x* \sum_{1 }^{ \infty } na_nx^{n-1} +\sum_{0 }^{ \infty }a_nx^n$

I have to rewrite this expression with one just involving $\displaystyle x^n$ apparently the answer is $\displaystyle \sum_{0 }^{ \infty }(n+1)a_nx^n$ but i dont know how to combine the two series.. any help?

2. So if the limits of your sum are the same and are over the same parameter then you can just put them together:

Since you are not summing over x you can just put it inside:
$\displaystyle x* \sum_{1 }^{ \infty } na_nx^{n-1} +\sum_{0 }^{ \infty }a_nx^n = \sum_{1 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n$

Then you can change the limits of the first sum if you subtract the zero term afterwards:

$\displaystyle \sum_{1 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n = \sum_{0 }^{ \infty } na_nx^{n} - 0a_0x^{0}+\sum_{0 }^{ \infty }a_nx^n = \sum_{0 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n$

Now we have the same limits over the same parameter so we can just put them together:

$\displaystyle \sum_{0 }^{ \infty } na_nx^{n} +\sum_{0 }^{ \infty }a_nx^n = \sum_{0 }^{ \infty } na_nx^{n} + a_nx^n = \sum_{0 }^{ \infty } a_nx^{n}(n+1)$