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What is an identity for $$\sum_{n=1}^\infty \frac{1}{k^n}\quad ?$$ I found numerous identities for $\sum_{n=1}^\infty ak^n$, all of which extremely complex, but are there any simpler identities?

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not a good edit, who is m? –  user01123581321345589144... Apr 19 '13 at 21:46
    
@user67133 Fixed, thanks :) –  KevinOrr Apr 19 '13 at 21:47
    
the usage of k,n in the summation was not conventional. I modified to what I think should be the more conventional way, please review. –  Arjang Apr 19 '13 at 22:50
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@Arjang: since there is an answer already using the original notation, let's leave it. –  robjohn Apr 19 '13 at 23:13
    
@KevinOrr : Google geometric series –  Stefan Smith Apr 19 '13 at 23:18
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1 Answer 1

up vote 6 down vote accepted

The formula is extremely simple: when $|k|<1$, $$\sum_{n=1}^\infty ak^n=\frac{ak}{1-k}.$$ In particular, if you let $\ell=\frac{1}{k}$, then $$\sum_{n=1}^\infty\frac{1}{k^n}=\sum_{n=1}^\infty\ell^n,$$ and now apply the first identity. Note that you need $|\ell|<1$ in order for the first identity to hold, and that this is equivalent to $|k|>1$.

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You've mixed up all the variables so that they have (1) nothing to do with the original question, (2) aren't consistent between the two equations in your answer. –  Thomas Andrews Apr 19 '13 at 22:52
    
@Thomas: Ahem, Arjang did that when he edited the question. –  Zev Chonoles Apr 19 '13 at 22:53
    
@Zev : Do you think n should be used for upper limit and k as the index of summation? seeing k and usage like this is unfamiliar. Of course the answer is the same either way but it looks a bit odd –  Arjang Apr 19 '13 at 22:53
    
@Zev : sorry, was just writig to you about the edit. –  Arjang Apr 19 '13 at 22:54
    
@ThomasAndrews : My fault, apologies, I was confused by the usage of n and k, tried to edit to something more conventional ( I think). –  Arjang Apr 19 '13 at 22:56
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