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Determine if the series converges absolutely, converges conditionally, or diverges. Find the exact value for the sum of the convergent series. $$1-\frac{1}{5} - \frac{1}{5^2} + \frac{1}{5^3} - \frac{1}{5^4} - \frac{1}{5^5} + \frac{1}{5^6} - \frac{1}{5^7} - \frac{1}{5^8} ...$$ I have no clue where to start. I tried using the comparison test comparing this series to the series to the series $\sum_{n=1}^{\infty}\frac{1}{n^2}= 2$ but that only tells me that this is a convergent series, not what value it converges to or if it is absolutely convergent or conditional convergent. Any advice and tips on how to solve this problem and these types of in general would be greatly appreciated. Thank you in advance :)

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This converges absolutely if you compare with the geometric series with $r = 1/5$. More specifically, it is less than $$ \sum_{i \ge 0} (1/5)^i = \frac {1}{1-1/5} $$

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  • $\begingroup$ Wow. Cannot believe I hadn't considered the geometric series. It's been a long night. Huge thank you to everyone who replied $\endgroup$
    – Todd Jones
    Oct 17, 2021 at 3:48
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    $\begingroup$ @ToddJones To find the value of the series, note that you original series is also a geometric series, with $r=-1/5$ (and with the whole series multipied by $-1$). $\endgroup$
    – Clement C.
    Oct 17, 2021 at 3:56
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    $\begingroup$ @Clement: I don't think that is quite correct. The series always has one positive and two negative terms. So I guess one needs to subtract two geometric series from each other (or something similar). $\endgroup$
    – PhoemueX
    Oct 17, 2021 at 5:40
  • $\begingroup$ @PhoemueX ouch, my bad, I misread the OP's series. I thought the signs were alternating... $\endgroup$
    – Clement C.
    Oct 17, 2021 at 6:33

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