summation from n = 0 -> infinity of:
$\displaystyle \sum\frac{n}{2^n}x^n$
how do you find it? completely lost.
I tried divergence and convergence stuff but it doesnt seem to net me the correct answer.
use the ratio test to find the radius of convergence ...
$\displaystyle \lim_{n \to \infty} \left|\frac{(n+1)x^{n+1}}{2^{n+1}} \cdot \frac{2^n}{n x^n} \right| < 1$
$\displaystyle \lim_{n \to \infty} \left|\frac{(n+1)x}{2n}\right| < 1$
$\displaystyle \left|\frac{x}{2}\right| \lim_{n \to \infty} \frac{n+1}{n} < 1$
$\displaystyle \left|\frac{x}{2}\right| \cdot 1 < 1$
$\displaystyle -1 < \frac{x}{2} < 1$
$\displaystyle -2 < x < 2$