∑ ∞ n = 1 − 4 ( − 1 2 ) n −1

Question

Does the series converge or diverge? If it converges, what is the sum? Show your work

Answer 1

I read this as ∑{n=[1,∞) }-4(-½)ⁿ⁻¹, in words:

the sum between n=1 and infinity of negative 4 times negative one half to the power of n-1.

If we use S to represent the sum, then S=-(4-2+1-½+¼-...)=-(2+1-½+¼-⅛+...).

Let G=1-½+¼-⅛+... =1/(1-(-½)) a convergent geometric progression.

G=1/(3/2)=⅔.

S=-2-G=-2⅔=-8/3.