Question

Does the following infinite series converge or diverge?


`(1)/(2) - (2)/(9) + (4)/(27) + - (8)/(81) +`
A. It Diverges; it does not have a sum

B. It converges; it does not have a sum

C. It Diverges; it has a sum

D. It converges; it has a sum

Answer 1

The nth term is (presuming the denominator on the first is a typo)


`-(n)/((-3)^n)`

The ratio of successive terms is


`( (n+1)/((-3)^(n+1)))/((n)/((-3)^n)) = (n+1)/(-3n)`

The limit is -1/3, absolute value less than 1, so the series converges. It has a sum -- that's what converges means.