Determine if ∑ n=2 to ∞ 1/ (n *(ln(n))^(1/2)) is convergent or divergent using Integral Test.

Question

Determine if ∑ n=2 to ∞ 1/ (n *(ln(n))^(1/2)) is convergent or divergent using Integral Test.

Answer 1

If we integrate 1/(n (ln n)^(1/2)), we get (by using substitution) 2(ln n)^(1/2). Clearly, the limit as n goes to infinity diverges (upper bound of integral). So, the integral test tells us that because the integral diverges, the given infinite sum also diverges.