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## Homework Statement

Well, the original question is to solve this ...

[tex]\sum[/tex] 1/(a

^{2}+ x

^{2})

the sum goes from x=-infinity to infinity (i wasnt sure how to show this with the latex??)

and the answer i am supposed to show is [tex]\pi[/tex]/a + (2*[tex]\pi[/tex]/a) * (1/(e

^{2*[tex]\pi[/tex]*a}- 1)

## Homework Equations

So, using Poisson resummation and some results from previous exercises, i get the new problem to being

[tex]\sum[/tex] ([tex]\pi[/tex]/a) * (e

^{-2*[tex]\pi[/tex]*a*|v|})

the sum now from v=-infinity to infinity

## The Attempt at a Solution

so i can split this into three.

the easy bit is to take the v=o part, this gives me the [tex]\pi[/tex]/a required in the answer.

then i have two parts, the sum with v=1 to infinity and the sum with v=-infinity to -1

these two parts are identical, due to the |v|.

so i have 2*[tex]\sum[/tex] ([tex]\pi[/tex]/a) * (e

^{-2*[tex]\pi[/tex]*a*|v|})

i can take the 2*[tex]\pi[/tex]/a outside of the sum, as it has no v element.

so now i am left having to show that

[tex]\sum[/tex] (e

^{-2*[tex]\pi[/tex]*a*|v|}) = (1/(e

^{2*[tex]\pi[/tex]*a}- 1)

which, i realise is just a (i think) simple geometric series sum ... and yet im not sure how to show these two equate?

any help would be brilliant, thank you.