Question

∞∑n=1 -4(-1/2) n-1

Answer 1

I'll assume you mean

∞
∑ -4(-1/2)^(n-1)
n=1

That sum is just
-4*∑(-1/2)^(n-1)

which is a geometric progression with
a = 1
r = -1/2
and sum = 1/(1-r) = 2/3

So, your sum is just -8/3