Determine whether the geometric series ∑∞n=1 (4^(n−1))/9^n is convergent or divergent. If it is convergent, find its sum. Give reasons and show all your work.

Answer 1

Examine the series

1/9 + 4/9² + 4²/9³

Since the common ratio, 4/9, is less than 1 the series converges.

Answer 2

Your sequence is equal to 1/9 ∑ (4/9)ⁿ summed from 0 to ∞.

geometric series Σ rⁿ , n = 0 to ∞, converges to 1/(1-r) for all r < 1

Your series converges to 1/9/(1-4/9)

1/9 * 9/5 = 1/5

Answer 3

If the terms of the series are 4^n-1/9n, then

the series cannot converge because lim (as n->∞) 4^n-1/9n is not 0.

If the terms are something different, then all bets are off!