# Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 556: 30

Converges

#### Work Step by Step

Given $$\sum_{n=1}^{\infty}\frac{n !}{n^{3}}$$ Compare with $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{2}}$, which is a convergent series ( $p-$series with $p=2$) and for $n\geq 1$ \begin{align*} \frac{n !}{n^{3}}&=\frac{n \times(n-1) !}{n^{3}}\\ \frac{n !}{n^{3}}&=\frac{(n-1) !}{n^{2}}\\ & \geq \frac{1}{n^2} \end{align*} Then $\displaystyle\sum_{n=1}^{\infty} \frac{n !}{n^{3}}$ also converges.

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