Q20P Use the ratio test to find wheth... [FREE SOLUTION]

Chapter 1: Q20P (page 1)

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

Short Answer

Expert verified

The series ${鈭�}_{n = 0}^{鈭�} \frac{n!}{(2 n)!}$ is convergent.

Step by step solution

Step 1: Process of ratio test

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

Apply the ratio test

The given series is ${鈭�}_{n = 0}^{鈭�} \frac{n!}{(2 n)!}$ .

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

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

$\begin{array}{rcl} 蚁_{n} & = & |\frac{a_{n + 1}}{a_{n}}| \\ = & |\frac{(n + 1)!}{[2 (n + 1)]!} 梅 \frac{n!}{2 n!}| \\ = & |\frac{(n + 1)!}{[2 (n + 1)]!} 脳 \frac{2 n!}{n!}| \\ = & \frac{(n + 1)}{(2 n + 2) (2 n + 1)} \\ = & \frac{1}{2 (2 n + 1)} \end{array}$

Solve the limit

Now, $蚁 = \underset{n 鈫� 鈭�}{l i m} 蚁_{n}$ is calculated as follows:

role="math" $\begin{array}{rcl} 蚁 & = & \underset{n 鈫� 鈭�}{l i m} \frac{1}{2 (2 n + 1)} \\ = & \frac{1}{2 (鈭� + 1)} \\ = & \frac{1}{鈭�} \\ 蚁 & = & 0 \end{array}$

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

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91影视

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

Short Answer

Step by step solution

Step 1: Process of ratio test

Apply the ratio test

Solve the limit

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91影视

Use the ratio test to find whether the following series converge or diverge:20.鈭�n=0鈭�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

Further Mechanics and Thermal Physics

Medical Physics

Torque and Rotational Motion

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Study anywhere. Anytime. Across all devices.

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