Does the series converge or diverge? If it converges, what is the sum? ∞Σ n=1 -4(-1/3)^n-1

Answer 1

To determine if the series converges, we can rewrite it in a more simplified form:

∞Σ n=1 -4(-1/3)^n-1
= -4 * ∞Σ n=1 (-1/3)^n-1
= -4 * ∞Σ n=0 (-1/3)^n

This is a geometric series with first term a = 1 and common ratio r = -1/3. The series converges if |r| < 1, so |r| = |-1/3| = 1/3 < 1. Therefore, the series converges.

The sum of the series can be found using the formula for the sum of an infinite geometric series:

Sum = a / (1 - r)
Sum = 1 / (1 + 1/3)
Sum = 1 / (4/3)
Sum = 3/4

So, the series converges and the sum is 3/4.