Interval of convergence of a power series, with a check for convergence at endpoints

Question

Find the interval of convergence of the power series. Be sure to include a check for convergence at the endpoints of the interval. $$(a) \ \sum_{n=1}^\infty \frac{(-1)^n x^n}{n} \qquad (b) \ \sum_{n=1}^\infty \frac{(-1)^{n+1}(x-5)^n}{n\cdot 5^n}$$

I know you use the power series and start with ratio test , test the points at the end. Here is my work so far. Does it look like I am doing the right thing for B? (The solution for A is in an answer)

Answer 1

At this point you can factor $x$ from your expression to obtain

$\displaystyle |x| \lim_{n\rightarrow \infty}\left| \frac{n}{n+1}\right|$. You should be able to find this limiting value.

At that point, you know that the series converges when this expression is less than 1.

Answer 2

I believe this is the answer for the first problem. Please correct me if I am wrong