Use Theorem 9.11 to determine the convergence or divergence of the series.
A)diverges
B)converges
C)Theorem 9.11 inconclusive

Question

Use Theorem 9.11 to determine the convergence or divergence of the series. ​  ​
A)diverges 
B)converges 
C)Theorem 9.11 inconclusive

Quizplus · Accepted Answer

Theorem 9.11 states that if the limit of the absolute value of the terms of a series is zero, then the series converges. In this case, we can use the limit comparison test with the series 1/n^3.

We have:

lim (n → ∞) (11/n^3) / (1/n^3) 
= lim (n → ∞) 11 
= 11

Since this limit is finite and positive, we conclude that the series 11/n^3 converges if and only if the series 1/n^3 converges. But the series 1/n^3 is a known convergent p-series with p = 3 > 1.

Therefore, by the limit comparison test, we conclude that the series 11/n^3 also converges. Thus, the answer is B.