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