I am struggling to find the sum of the following series:
$$\sum_{n=1} ^{\infty} \frac{(-1)^n}{(n+1)(n+3)(n+5)}.$$
It seems as though it should be a straightforward telescoping series. I attempted to solve it in the usual way (via partial fractions), but the alternating sign makes the sum so that one cannot cancel out fractions to result in a finite sum of fractions. I know that the series converges by the alternating sign test, and I check on WolframAlpha that the infinite sum converges to $-7/480$. Any thoughts on how to proceed?