Match the following series with the series below in which you can compare using the Limit Comparison Test. Then determine whether the series converge or diverge.

Question

Match the following series with the series below in which you can compare using the Limit Comparison Test. Then determine whether the series converge or diverge.

A. ∑(1/n),n=1 to infinity, B. ∑(1/n^(2)),n=1 to infinity, C. ∑(1/n^(3)),n=1 to infinity, and D. ∑(1/n^(3/2),n=1 to infinity

 

1. ∑1/(sqrt(n^3+1)),n=1 to infinity Does this series converge or diverge?

2. ∑(n+2)/((n+1)^3),n=1 to infinity Does this series converge or diverge?

3. ∑1/(2+n^(3/2)),n=1 to infinity Does this series converge or diverge?

4. ∑(n^2-1)/(n^4+2n+1),n=1 to infinity Does this series converge or diverge?

Answer 1

First of all, of the comparison series given: A diverges and the others converge.

All of the series given converge and there is a general result which will give you the answer directly: the integral test shows that the infinite series with terms 1/n^p converges for p>1 and diverges for p≤1.

However, the limit comparison test will work as well, as long as you know about the convergence of the series A through D.