which series does the ratio test fail to give a conclusive answer. options in comments

Question

1. from n=1 to infinity (1)/(n^3)
2. from n=1 to infinity (n)/(2^n)
3. from n=1 to infinity (square root n)/(1+n^2)

Answer 1

1) lim n³/(n+1)³ = 1

2) lim ((n+1) 2²)/((2ⁿ⁺¹)n) = 1/2 < 1

3) lim √((n+1)/n) (1+(n+1)²)/(1+n²) = 1

The limit must be >1 or <1 to make a determination. So both Series (1) and (3) fail the root test.

Answer 2

The answer is choices 1 and 3

Reason for choice 1: If we divide the (n+1)st term by the nth term and simplify we get

n³ / (n+1)³ = (n/(n+1))³ = (1 / (1+1/n))³.

The limit of the expression above as n approaches infinity is 1, which tells us that the Ratio Test is inconclusive..

For choice 2, the limit as n approaches infinity of the (n+1)st term divided by the nth term is 1/2. So, by the Ratio Test, the series converges.

For choice 3, if we divide the (n+1)st term by the nth term and simplify we get

√(n/(n+1)))[(n² + 2n + 2) / (n² + 1)] which has limit equal to 1 as n goes to infinity. So, the Ratio Test is inconclusive.