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

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)

• So far i tried the ratio test. I posted my results – Zak Dec 11 '14 at 0:27
• Does it look like I am doing the right thing for B? Thanks – Zak Dec 11 '14 at 1:32
• When you factor $-1/5$ out of absolute values, you should be taking the absolute value of this, making it $\frac{1}{5}|x-5|<1$. – Addem Dec 11 '14 at 2:14

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.
• Your solution for (a) is nearly correct. You correctly compute the limit in the Ratio Test and give the correct bounds on $x$. For when $x=-1$ you don't get an alternating series, though, since $(-1)^{n}\cdot (-1)^{n} = (-1)^{2n} = 1$ for all $n$. So this is just the harmonic series which doesn't converge. It is the alternating series you give when $x=1$ though, so it converges there. – Addem Dec 11 '14 at 2:12