Chapter 1: Q22P (page 1)

Use the ratio test to find whether the following series converge or diverge:
22. $\sum_{n = 1}^{\infty} \frac{10^{n}}{{(n!)}^{2}}$

Short Answer

Expert verified

The series $\sum_{n = 1}^{\infty} \frac{10^{n}}{{(n!)}^{2}}$ is convergent.

Step by step solution

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 ${(n + 1)}^{t h}$ term of the series and $a_{n}$ is the $n^{t h}$ term. If role="math" $ρ < 1$ , then the series converges. If role="math" $ρ > 1$ , then the series diverges.

Apply the ratio test

The given series is $\sum_{n = 1}^{\infty} \frac{10^{n}}{{(n!)}^{2}}$ .

So, $a_{n + 1} = \frac{10^{n + 1}}{{[(n + 1)!]}^{2}}$ , and localid="1664185894811" $a_{n} = \frac{10^{n}}{{(n!)}^{2}}$ .

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

$\begin{array}{rcl} ρ_{n} & = & |\frac{a_{n + 1}}{a_{n}}| \\ = & |\frac{10^{n + 1}}{{[(n + 1)!]}^{2}} \div \frac{10^{n}}{{(n!)}^{2}}| \\ = & \frac{10^{n + 1}}{{[(n + 1)!]}^{2}} \times \frac{{(n!)}^{2}}{10^{n}} \\ = & \frac{10}{{(n + 1)}^{2}} \end{array}$

Solve the limit

Now, $ρ = \lim_{n \to \infty} ρ_{n}$ is calculated as follows:

role="math" localid="1664186203147" $\begin{array}{rcl} ρ & = & \lim_{n \to \infty} \frac{10}{{(n + 1)}^{2}} \\ = & \frac{10}{{(\infty + 1)}^{2}} \\ = & \frac{10}{\infty} \\ = & 0 \end{array}$

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

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Use the ratio test to find whether the following series converge or diverge:
22. $\sum_{n = 1}^{\infty} \frac{10^{n}}{{(n!)}^{2}}$

Short Answer

Step by step solution

Process of ratio test

Apply the ratio test

Solve the limit

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Use the ratio test to find whether the following series converge or diverge:22.∑n=1∞10n(n!)2

Short Answer

Step by step solution

Process of ratio test

Apply the ratio test

Solve the limit

One App. One Place for Learning.

Most popular questions from this chapter

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Use the ratio test to find whether the following series converge or diverge:
22. $\sum_{n = 1}^{\infty} \frac{10^{n}}{{(n!)}^{2}}$