Chapter 9 - Section 9.3 - The Integral Test - Exercises - Page 504: 27

Diverges

Work Step by Step

Consider $a_n=\lim\limits_{n \to \infty} \dfrac{\sqrt n}{\ln n}$ But $\lim\limits_{n \to \infty} \dfrac{\sqrt n}{\ln n}=\dfrac{\infty}{\infty}$ Need to apply L-Hospital's rule. $\lim\limits_{n \to \infty} \dfrac{1/2 \sqrt n}{1/n} =\lim\limits_{n \to \infty} \dfrac{\sqrt n}{2}=\infty$ Hence, the given series diverges.

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