Ratio Test

Examples

Determine if the following series are convergent:

$\sum_{n = 1}^{\infty} \frac{5 n}{n!}$

\begin{aligned} lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | & = lim_{n \to \infty} | \frac{5 (n + 1)}{(n + 1)!} \frac{n!}{5 n} | \\ = lim_{n \to \infty} | \frac{n!}{(n + 1)!} \frac{5 (n + 1)}{5 n} | \\ = lim_{n \to \infty} | \frac{n + 1}{(n + 1) n} | \\ = 0 < 1 \end{aligned}

Therefore, $\sum a_{n}$ is convergent by RT.

$\sum_{n = 1}^{\infty} \frac{2^{n} n!}{(3 n) (2 n)!}$

\begin{aligned} lim_{n \to \infty} | \frac{2^{n + 1} (n + 1)!}{(3 (n + 1)) (2 (n + 1))!} \frac{(3 n) (2 n)!}{2^{n} n!} | & = lim_{n \to \infty} | \frac{2^{n + 1}}{2^{n}} \frac{(n + 1)!}{n!} \frac{3 n}{3 n + 3} \frac{(2 n)!}{(2 n + 2)!} | \\ = lim_{n \to \infty} | \frac{2 (n + 1) (3 n)}{(3 n + 3) (2 n + 2) (2 n + 1)} | \\ = 0 < 1 \end{aligned}

Therefore, $\sum a_{n}$ is convergent by RT.

$\sum_{n = 1}^{\infty} \frac{4 (n!)^{2}}{(2 n)!}$

\begin{aligned} lim_{n \to \infty} | \frac{4 ((n + 1)!)^{2}}{(2 (n + 1))!} \frac{(2 n)!}{4 (n!)^{2}} | & = lim_{n \to \infty} | \frac{4 ((n + 1)!)^{2}}{4 (n!)^{2}} \frac{(2 n)!}{(2 n + 2)!} | \\ = lim_{n \to \infty} | \frac{(n + 1)^{2}}{(2 n + 2) (2 n + 1)} | \\ = \frac{1}{4} < 1 \end{aligned}

Therefore, $\sum a_{n}$ is convergent by RT.

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