Find all the real values of $x$ for which $\sum_1^{\infty}(x^n/n)$ converges.
I began with the ratio test to get $nx/(n+1)$ but I'm not sure where to go next. I think I'm supposed to use Leibniz's Theorem at some point?
Find all the real values of $x$ for which $\sum_1^{\infty}(x^n/n)$ converges.
I began with the ratio test to get $nx/(n+1)$ but I'm not sure where to go next. I think I'm supposed to use Leibniz's Theorem at some point?
$a_n:=\frac{x^n}{n}\Longrightarrow \left|\frac{a_{n+1}}{a_n}\right|=\left|\frac{x^{n+1}}{n+1}\frac{n}{x^n}\right|=|x|\frac{n}{n+1}\xrightarrow [n\to\infty]{}|x|$
Thus, the series converges for $\,-1