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

Question

∑∞n=1 -4(-1/2)^n-1

Answer 1

This series is a geometric series with first term a=-4 and common ratio r=-1/2.

For a geometric series to converge, the absolute value of the common ratio must be less than 1. |r|<1

|-1/2|<1

Therefore, the series converges.

To find the sum of a convergent geometric series, use the formula:

sum = a / (1 - r)

where a is the first term and r is the common ratio.

In this case, a = -4 and r = -1/2.

sum = (-4) / (1 - (-1/2))

sum = (-4) / (3/2)

sum = -8/3

Therefore, the sum of the series is -8/3.