Does the series converge or diverge? If it converges, what is the sum? Show your work. infinity ∑ n=1 - 4 (-1/2) ^n-1

It converges because the common ratio, r, is 1/4. Any time r^2<1 the series converges and thus a sum can be calculated.

The sum of any geometric sequence can be expressed as:

s (n) = a (1-r^n) / (1-r), and as a result, whenever r^2<1 this sum is:

s=a / (1-r)

In this case a=-4 and r=-0.5 so

s=-4 / (1--0.5)

s=-4 / (1.5)

s=-2 2/3