Chapter 1: Q10P (page 1)

Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.
10. $\sum_{n = 1}^{\infty} \frac{e^{n}}{e^{2 n} + 9}$

Short Answer

Expert verified

The series $\sum_{n = 1}^{\infty} \frac{e^{n}}{e^{2 n} + 9}$ is convergent.

Step by step solution

Definition of convergent and divergent

If the partial sum $S_{n}$ of an infinite series tend to a limit S, then the series is called convergent.

If the partial sum $S_{n}$ of an infinite series do not approach to a limit, the series is called divergent.

The limiting value S is called the sum of the series.

Apply the integral test

The given series is $\sum_{n = 1}^{\infty} \frac{e^{n}}{e^{2 n} + 9}$ .

Use the integral in the given series as:

$\int_{}^{\infty} \frac{e^{n}}{e^{2 n} + 9} d n$

Solve the integral as follows:

Let, $t = e^{n}$ , so $d t = e^{n} d n$ .

Solve the integral

Substitute the value of t and dt into the integral and change the variable from n into t.

$\int_{}^{\infty} \frac{1}{t^{2} + 9} d t = \int_{}^{\infty} \frac{1}{t^{2} + 3^{2}} d t$

Use the integral formula $\int \frac{1}{x^{2} + a^{2}} d x = \frac{1}{a} \tan^{- 1} (\frac{x}{a}) + c$ .

$\begin{array}{rcl} \int_{}^{\infty} \frac{1}{t^{2} + 9} d t & = & \frac{1}{3} {[\tan^{- 1} (\frac{t}{3})]}^{\infty} \\ = & \frac{1}{3} [\tan^{- 1} (\frac{\infty}{3})] \\ = & \frac{1}{3} [\tan^{- 1} (\infty)] \\ = & \frac{1}{3} [\frac{π}{2}] \\ = & \frac{π}{6} \end{array}$

Here, the series approaches to a finite value, therefore the given series converges.

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Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.
10. $\sum_{n = 1}^{\infty} \frac{e^{n}}{e^{2 n} + 9}$

Short Answer

Step by step solution

Definition of convergent and divergent

Apply the integral test

Solve the integral

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Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.10.∑n=1∞ene2n+9

Short Answer

Step by step solution

Definition of convergent and divergent

Apply the integral test

Solve the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Scientific Method Physics

Electricity

Fields in Physics

Modern Physics

Linear Momentum

Study anywhere. Anytime. Across all devices.

Use the integral test to find whether the following series converge or diverge. Hint and warning: Do not use lower limits on your integrals.
10. $\sum_{n = 1}^{\infty} \frac{e^{n}}{e^{2 n} + 9}$