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Say I want to find the value that the series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ converges to. We would use the infinite geometric sum formula $S_\infty=\frac{a}{1-r}=2$

And this works for all $|r|<1$.

Now say we use the same formula on $1+2+4+8+16+...$

We get the nonsensical answer $\frac{1}{1-2}=-1$

Is there any actual significance of using the infinite geometric sum formula for when $|r|>1$ ?

Trogdor
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    No. But using the $2$-adic topology you have $\lvert 2\rvert_2=1/2<1$ hence the series $\sum_{n\geq0}2^n$ converges to $-1$ in the $2$-adic sense. – gniourf_gniourf Jun 28 '14 at 15:32
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    Have a look at this answer. – Start wearing purple Jun 28 '14 at 15:38
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    For the first sum you can say "if this sum has a finite limit S, then S = 1 + S/2". For the second sum you can say "if this sum has a finite limit S, then S = 1 + 2S". Both statements are correct. It follows that the first sum cannot have a finite limit other than 2, the second cannot have a finite limit other than -1. Unfortunately, the second sum has no finite limit at all :-) – gnasher729 Jun 28 '14 at 19:18

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