Answer:
This series is convergent. The partial sums of this series converge to 
.
Step-by-step explanation:
The 
th partial sum of a series is the sum of its first 
terms. In symbols, if 
denote the 
th term of the original series, the 
th partial sum of this series would be:
.
A series is convergent if the limit of its partial sums, 
, exists (should be a finite number.)
In this question, the th term of this original series is:
.
The first thing to notice is the 
in the expression for the th term of this series. Because of this expression, signs of consecutive terms of this series would alternate between positive and negative. This series is considered an alternating series.
One useful property of alternating series is that it would be relatively easy to find out if the series is convergent (in other words, whether exists.)
If 
is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit exists) as long as 
.
For the alternating series in this question, indeed:
.
Therefore, this series is indeed convergent. However, this conclusion doesn't give the exact value of . The exact value of that limit needs to be found in other ways.
Notice that is a geometric series with the first term is 
while the common ratio is 
. Apply the formula for the sum of geometric series to find an expression for 
:
.
Evaluate the limit :
![\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20%5Clim%5Climits_%7Bn%20%5Cto%20%5Cinfty%7D%20S_%7Bn%7D%20%26%3D%20%5Clim%5Climits_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cleft%28-%5Cfrac%7B2%7D%7B3%7D%20%2B%20%5Cfrac%7B2%7D%7B3%7D%20%5Ccdot%20%7B%5Cleft%28-%5Cfrac%7B1%7D%7B2%7D%5Cright%29%7D%5E%7Bn%7D%5Cright%29%20%5C%5C%20%26%3D%20-%5Cfrac%7B2%7D%7B3%7D%20%2B%20%5Cfrac%7B2%7D%7B3%7D%20%5Ccdot%20%5Cunderbrace%7B%5Clim%5Climits_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cleft%5B%7B%5Cleft%28-%5Cfrac%7B1%7D%7B2%7D%5Cright%29%7D%5E%7Bn%7D%20%5Cright%5D%20%7D_%7B0%7D%3D%20-%5Cfrac%7B2%7D%7B3%7D%5Cend%7Baligned%7D%7D_)
.
Therefore, the partial sum of this series converges to 
.
