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Conclude if the given series converges or diverges and justify the conclusion

by RoRi

Test the following series for convergence or divergence. Justify the decision.

    \[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n(n+1)}}. \]

The series diverges. We apply the limit ratio test with

    \[ a_n = \frac{1}{\sqrt{n(n+1)}}, \qquad b_n = \frac{1}{n}. \]

Then taking the limit of the ratio we have

    \begin{align*} \lim_{n \to \infty} \frac{a_n}{b_n} &= \lim_{n \to \infty} \frac{\frac{1}{\sqrt{n^2+n}}}{\frac{1}{n}} \\[9pt] &= \lim_{n \to \infty} \frac{n}{\sqrt{n^2+n}} \\[9pt] &= \lim_{n \to \infty} \frac{1}{\sqrt{1+\frac{1}{n}}} \\[9pt] &= 1. \end{align*}

Thus, both \sum a_n and \sum b_n converge or both diverge. But, we know \sum b_n = \sum \frac{1}{n} diverges. Hence, \sum a_n diverges as well.

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