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

One may write
$$
\sum_{-\infty}^na^k=\sum_{-\infty}^0 a^k+\sum_1^n a^k
$$ then one may use the standard geometric series
$$
\sum_{k=0}^n x^k=\frac{1-x^{n+1}}{1-x}, \,\, x\neq1,\quad \sum_{k=0}^\infty x^k=\frac{1}{1-x},\quad |x|<1.
$$
Finally the sought sum is

$$
\sum_{-\infty}^na^k=\sum_0^{\infty} a^{-k}+\sum_1^n a^k=\frac{1}{1-\frac1a}+\frac{a(1-a^{n})}{1-a}=\frac{a^{n+1}}{a-1},\quad |a|>1.
$$

