## anonymous 5 years ago Determine a function which is represented by that summation given below

1. anonymous

$\sum _{n=1}^{\infty } \text{nx}^n$

2. anonymous

ok i guess we will worry about the radius of convergence second. this looks almost like a derivative, so we will treat it as one.

3. anonymous

$\Sigma_1^\infty x^n=\frac{1}{1-x}$ for -1<x<1 yes?

4. anonymous

yes

5. anonymous

taking derivatives we get $\Sigma_1^\infty nx^{n-1}=\frac{1}{(1-x)^2}$

6. anonymous

okay

7. anonymous

multiply both sides by x to get $\Sigma_1^\infty nx^n=\frac{x}{(1-x)^2}$

8. anonymous

i have been very sloppy here, especially from where we started. i think we have to be careful with starting at n = 0 or n = 1, but i will let you worry about this. i will also let you worry about the validity of differentiating a power series term by term and the radius of convergence. but the general idea is there and i think the answer is correct modulo those details.

9. anonymous

Okay,thanks

10. anonymous

i think in fact the very first line i wrote was incorrect. i think $\Sigma_0^\infty x^n = \frac{1}{1-x}$ no what i wrote.

11. anonymous

but of course when n = 0 in your power series you get 0 anyway, so you might as well start at 1!