Chapter 1: Q20P (page 1)

Use the ratio test to find whether the following series converge or diverge:
20. $\sum_{n = 0}^{\infty} \frac{n!}{(2 n)!}$

Short Answer

Expert verified

The series $\sum_{n = 0}^{\infty} \frac{n!}{(2 n)!}$ is convergent.

Step by step solution

Step 1: Process of ratio test

Apply ratio test in the given series by using $ρ_{n} = | \frac{a_{n + 1}}{a_{n}} |$ and $ρ = \lim_{n \to \infty} ρ_{n}$ , where $a_{n + 1}$ is the ${(n + 1)}^{th}$ term of the series and $a_{n}$ is the $n^{th}$ term. If role="math" $ρ < 1$ , then the series converges. If $ρ > 1$ , then the series diverges.

Apply the ratio test

The given series is $\sum_{n = 0}^{\infty} \frac{n!}{(2 n)!}$ .

So, $a_{n + 1} = \frac{(n + 1)!}{[2 (n + 1)]!}$ , $a_{n} = \frac{n!}{2 n!}$ .

Obtain the value of $ρ_{n} = |\frac{a_{n + 1}}{a_{n}}|$

$\begin{array}{rcl} ρ_{n} & = & |\frac{a_{n + 1}}{a_{n}}| \\ = & |\frac{(n + 1)!}{[2 (n + 1)]!} \div \frac{n!}{2 n!}| \\ = & |\frac{(n + 1)!}{[2 (n + 1)]!} \times \frac{2 n!}{n!}| \\ = & \frac{(n + 1)}{(2 n + 2) (2 n + 1)} \\ = & \frac{1}{2 (2 n + 1)} \end{array}$

Solve the limit

Now, $ρ = \underset{n \to \infty}{l i m} ρ_{n}$ is calculated as follows:

role="math" $\begin{array}{rcl} ρ & = & \underset{n \to \infty}{l i m} \frac{1}{2 (2 n + 1)} \\ = & \frac{1}{2 (\infty + 1)} \\ = & \frac{1}{\infty} \\ ρ & = & 0 \end{array}$

Here, $ρ < 1$ , therefore the series converges.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use the ratio test to find whether the following series converge or diverge:
20. $\sum_{n = 0}^{\infty} \frac{n!}{(2 n)!}$

Short Answer

Step by step solution

Step 1: Process of ratio test

Apply the ratio test

Solve the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Fluids

Translational Dynamics

Modern Physics

Fundamentals of Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Use the ratio test to find whether the following series converge or diverge:20.∑n=0∞n!(2n)!

Short Answer

Step by step solution

Step 1: Process of ratio test

Apply the ratio test

Solve the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Fluids

Translational Dynamics

Modern Physics

Fundamentals of Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Use the ratio test to find whether the following series converge or diverge:
20. $\sum_{n = 0}^{\infty} \frac{n!}{(2 n)!}$