TITLE: How to find the sum of an alternating series?
QUESTION [8 upvotes]: Find whether the series diverges and its sum:
  $$\sum_{n = 1}^\infty (-1)^{n+1} \frac{3}{5^n}.$$

I found that the series converges using the Alternating Series test because the absolute value of each $n$ decreases while the value of $n$ increases. Then I took the limit as $x$ approaches infinity of $3/5^x$ which is $0$. 
But I am not sure how to go about finding the sum at this point. This is a practice exam, so the solution is more important then the actual answer.

REPLY [5 votes]: Your series is an example of a geometric series. The first term is $a=3/5$, while each subsequent term is found by multiplying the previous term by the common ratio $r=-1/5$.
There is a well known formula for the sum to infinity of a geometric series with $|r| < 1$, namely:
$$S_{\infty} = \frac{a}{1-r} \, . $$
In your case, $a=3/5$ and $r=-1/5$, and so it follows that:
$$S_{\infty} =  \frac{3/5}{1+1/5} = \frac{1}{2} \, . $$