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Is it possible to make 2 geometric series from

$n = -i, -(i-1), -(i-2),...,(i-1),i$

For a finite series I know that a geometric series looks like that, for $x^n$

$ \sum_0^i = \frac{1-x^{i+1}}{1-x}$

happypaticle
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  • $$ \sum\limits_{n = - i}^i {x^n } = \sum\limits_{n = 1}^i {\left( {\frac{1}{x}} \right)^n } + \sum\limits_{n = 0}^i {x^n } $$ – Gary Nov 16 '22 at 04:07

1 Answers1

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Yes, it is possible. You have to be careful though.

You can split an expression like

$\sum_{k=-i}^i x^k$ into two sums.

$\sum_{k=0}^i x^k$ and $\sum_{k=-i}^0 x^k=\sum_{k=0}^i x^{-k}=\sum_{k=0}^i (1/x)^k$.

Where now the formula can also be applied to the second sum. Note that we count the summand for $k=0$ two times, so it has to be subtracted. Also note that for finite sums, the geometric series has not to fullfill the "convergence property" $|x|<1$, but can always be applied to sums of the above form.

Cornman
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