I have attempted to solve this problem:
Find the sum of the series, if it converges.
$\sum\limits_{n=1}^{\infty}\frac{(-2)^{n-1}}{7^n}$
I see that the values of $a_n$ are $\frac{1}{7} + \frac{-2}{49} + \frac{4}{343} + \frac{-8}{2401} + ...$
And that the values of $s_n$ are $\frac{1}{7},\frac{5}{49},\frac{39}{343},\frac{265}{2401},...$
I am trying to derive a formula for $s_n$ but feel stuck. I know the denominator of this mystery equation is $7^n$, but I can't seem to figure out what the rest of the equation would be. An explanation/corrections to my method would be greatly appreciated.
HINT:
$$\sum\limits_{n=1}^{\infty}\frac{(-2)^{n-1}}{7^n}=-\frac12\sum\limits_{n=1}^{\infty}\frac{(-2)^n}{7^n}=-\frac12\sum\limits_{n=1}^{\infty}\left(-\frac27\right)^n$$
Now see this