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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.

    \[ \sum_{n=1}^{\infty} \frac{1 + \sqrt{n}}{(n+1)^3 - 1}. \]

The series converges. We apply the limit ratio test, letting

    \[ a_n = \frac{1+\sqrt{n}}{(n+1)^3-1}, \qquad b_n = \frac{1}{n^{\frac{5}{2}}}. \]

Then, we know \sum b_n converges and we have

    \begin{align*} \lim_{n \to \infty} \frac{a_n}{b_n} &= \lim_{n \to \infty} \frac{\frac{1+\sqrt{n}}{(n+1)^3-1}}{\frac{1}{n^{\frac{5}{2}}}} \\[9pt] &= \lim_{n \to \infty} \frac{n^{\frac{5}{2}} + n^3}{n^3 + 3n^2 + 3n} \\[9pt] &= \lim_{n \to \infty} \frac{1 + \frac{1}{\sqrt{n}}}{1 + \frac{3}{n} + \frac{3}{n^2}} \\[9pt] &= 1. \end{align*}

Hence, the convergence of \sum b_n implies the convergence of

    \[ \sum_{n=1}^{\infty} \frac{1+\sqrt{n}}{(n+1)^3 - 1}. \]

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