Please show steps and explainwhat test did you use I will ra

Please show steps and explain....what test did you use? I will rate you

Determine the convergence or divergence of the following series, and state the test used:

Solution

Use the limit comparison test with 1/n^(1/2) as follows: lim(n->infinity) (n/sqrt(n^3 + 2n)) / (1/n^(1/2)) = lim(n->infinity) (n^(3/2))/(sqrt(n^3 + 2n)) = 1 (just use l\'hospital\'s rule) Therefore the series diverges since 1/n^(1/2) diverges by the p-series test.