# Thread: proving a series converges

1. ## proving a series converges

i have obtained the series for In(1+x) how can i find the radius of convergence? i have tried the ratio test but this doesn't seem to work for [(-1)^k* x^(k+1)]/k+1

2. $\displaystyle ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-...=\sum^{\infty}_{n=1} \frac{(-1)^{n+1}} {n}x^n$

Let us look on (*) $\displaystyle \sum^{\infty}_{n=1}| \frac{(-1)^{n+1}}{n}x^n|=\sum^{\infty}_{n=1} \frac{1}{n}|x^n|$

Using Cauchy test, (*) converges if $\displaystyle lim_{n\to\infty}\sqrt[n]{ \frac{1}{n}|x^n|}<1$

Or:

$\displaystyle |x|<1$

Now, check what happens when x=1 and x=-1.