Answer:
Step-by-step explanation:
Recall the ratio test. Given a series
if
![\lim_{n\to \infty} \left|\frac{a_{n+1}}{a_n}\right|](https://tex.z-dn.net/?f=%20%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20%5Cleft%7C%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%5Cright%7C%3C1)
Then, the series is absolutely convergent.
We will use this to the given series
, where
. Then, we want to find the values for which the series converges.
So
, which gives us that
![|2x|\cdot\lim_{n\to \infty} \frac{n}{n+1}](https://tex.z-dn.net/?f=%20%7C2x%7C%5Ccdot%5Clim_%7Bn%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bn%7D%7Bn%2B1%7D%3C1)
We have that
. Then, we have that
,
which implies that |x|<1/2. So for
the series converges absolutely.
We will replace x by the endpoints to check convergence.
Case 1, x=1/2:
In this case we have the following series:
which is the harmonic series, which is know to diverge.
Case 2, x=-1/2:
In this case we have the following series:
![\sum_{n=1}^{\infty} \frac{(-1)^n}{n}](https://tex.z-dn.net/?f=%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B%28-1%29%5En%7D%7Bn%7D)
This is an alternating series with
. Recall the alternating series test. If we have the following
and
meets the following criteria : bn is positive, bn is a decreasing sequence and it tends to zero as n tends to infinity, then the series converge.
Note that in this case,
si always positive, its' limit is zero as n tends to infinity and it is decreasing, hence the series converge.
So, the final interval of convergence is
![[\frac{-1}{2}, \frac{1}{2})](https://tex.z-dn.net/?f=%20%5B%5Cfrac%7B-1%7D%7B2%7D%2C%20%5Cfrac%7B1%7D%7B2%7D%29)