Geometric series from -i to i

Question

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}$

Answer 1

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.