Summation $ \sum_{k=-\infty}^n a^k$ Is there any formula that directly gives the result of
this summation:
$$
\sum_{k=-\infty}^n a^k$$
?

$$
S_{n+1}=\sum_{k=-\infty}^{n+1} a^k=S_n+a^{n+1}$$
and
$$S_{n+1}=\sum_{k=-\infty}^n a^{k+1}=aS_n$$ give
$$S_n=\frac{a^{n+1}}{a-1}.$$
