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Trouble calculating an alternating/infinite sum

Firstly, the sum is attached as a png file and the answer is that it converges (conditionally) to 1/12.

I thought about the fact that I could split that up to two sums yielding one with all the positive terms and one with all the negative terms and then evaluate the sums separately and then have simple arithmetic at the end but I don't know how to set up the positive/negative separation. In other words, I don't know what needs to be done mechanically in order to transform that one infinite sum into two infinite sums (one being positive and the other negative).

Any help would be greatly appreciated!

Thanks in advance!

Edit: If I am completely off in my reasoning, just please tell me how to do it in the first place.

Re: Trouble calculating an alternating/infinite sum

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Originally Posted by

**s3a** Firstly, the sum is attached as a png file and the answer is that it converges (conditionally) to 1/12.

I thought about the fact that I could split that up to two sums yielding one with all the positive terms and one with all the negative terms and then evaluate the sums separately and then have simple arithmetic at the end but I don't know how to set up the positive/negative separation. In other words, I don't know what needs to be done mechanically in order to transform that one infinite sum into two infinite sums (one being positive and the other negative).

Any help would be greatly appreciated!

Thanks in advance!

Edit: If I am completely off in my reasoning, just please tell me how to do it in the first place.

$\displaystyle \sum_{n=1}^{\infty} \frac{(-4)^{n-1}}{8^n} = -\frac{1}{4} \cdot \sum_{n=1}^{\infty} \frac{(-4)^{n}}{8^n} = -\frac{1}{4} \cdot \sum_{n=1}^{\infty} \left(-\frac{1}{2}\right)^n$

sum of an infinite geometric series ...

$\displaystyle S = -\frac{1}{4} \cdot \frac{-\frac{1}{2}}{1 - \left(-\frac{1}{2}\right)}$

Re: Trouble calculating an alternating/infinite sum