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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.

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3 Answers

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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 $n=1,2,3\dots$

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You're almost right but don't forget the alternating sign: the series is

$$\sum_{n=1}^\infty\frac{5(-1)^{n+1}}{n+8}$$

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By using $$\sum_{k=1}^\infty \frac{(-1)^k}{k}=-\ln 2,$$

$$\sum_{k=1}^\infty \frac{5(-1)^{n+1}}{n+8}=-5\sum_{k=9}^\infty \frac{(-1)^k}{n}=5\ln(2)+5\sum_{k=1}^8\frac{(-1)^k}{k}$$

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