I could use the help....

$\sum \limits_{n=0}^\infty \alpha^n = \dfrac{1}{1-\alpha},~\forall \alpha \ni |\alpha|<1\\ \text{and does not converge otherwise}$

$\sum \limits_{n=1}^\infty ~-4\left(-\dfrac 1 2\right)^{n-1} = -4\left(\sum \limits_{n=0}^\infty~\left(-\dfrac 1 2\right)^n \right)$

I leave it to you to apply the first line to the second.

List the first few terms

-4, 2, -1, 1/2, -1/4, 1/8........

The common ratio  is  -1/2    and the first term is -4

So....the sum converges to

   -4                    -4

_______  =    _____  =   -4 (2/3)  =  -8/3

1 - (-1/2)           (3/2)