I have calculated the Taylor series for $f(x)=\ln(1+x)$ centered at $x=0$ with some help and have gotten the following,
$$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}x^n$$
However now I am trying to find the values of $x$ for which this series converges. my thinking here was to use the ratio test,
$$\lim_{n\to\infty}\left|\frac{x^{n+1}}{n+1}\frac{n}{x^n}\right|=\lim_{n\to\infty}\left|\frac{xn}{n+1}\right|=|x|$$
so then by the ratio test it diverges when $x<-1,x>1$, and converges when $-1<x<1$
Looking at $-1,1$ the ratio test is inconclusive so, when $x=1$ it converges because the series is a convergent alternating series, however when $x=-1$ it diverges because it is just $-1$ multiplied by a divergent p series, as $p=1$
I was not sure if this logic is correct, I get messed up sometimes when the ratio test limit turns out to be $|x|$