[Math] Writing a general formula for an alternating series

Question

I need help for writing the general formula for following alternating series in the form

The alternating series are:

I feel that 5/(n+8) has something to do with this but I'm not sure how to make it alternate. Thanks in advance.

Answer 1

To make a series alternate, you generally stick in a factor of $(-1)^n$ or $(-1)^{n+1}$. In your case, the general term could be $(-1)^{n+1}\cdot \frac{5}{n+8}$ for $n=1,2,3\dots$