Q. 32 - Calculus

Question

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series 

$\sum _{k=1}^{\infty }\frac{k!}{\left(2k\right)!}$

Step-by-Step Solution

Answer

The series converges.

Step 1. Given information.

The given series is $\sum _{k=1}^{\infty }\frac{k!}{\left(2k\right)!}.$

Step 2. Root test.

According to the series,

$$\begin{gathered} \frac{a_{k+1}}{a_k}=\frac{\frac{(k+1)!}{(2k+2)!}}{\frac{k!}{\left(2k\right)!}} \\ =\frac{\left(2k\right)!(k+1)!}{k!(2k+2)!} \\ \frac{a_{k+1}}{a_k}=\frac{\left(2k\right)!(k+1)k!}{k!(2k+2)(2k+1)\left(2k\right)!} \\ =\frac{(k+1)}{2(k+1)(2k+1)} \\ =\frac{1}{2(2k+1)} \end{gathered}$$

Step 3. Conclusion.

On taking limits,

$$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}=\underset{k\to \infty }{\lim }\frac{1}{2(2k+1)} \\ =\frac{1}{2}\left(\underset{k\to \infty }{\lim }\frac{1}{2k+1}\right) \\ =0 \\ \text{ Since, }L<1, \end{gathered}$$

Therefore, the series converges.