Chapter 1: Q9P (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.
$\sum_{n = 3}^{\infty} \frac{1}{n^{2} - 4}$

Short Answer

Expert verified

The series $\sum_{n = 3}^{\infty} \frac{1}{n^{2} - 4}$ is divergent.

Step by step solution

Definition of convergent and divergent.

If the partial sums $S_{n}$ of an infinite series tend to a limit S, the series is called convergent. If the partial sums $S_{n}$ of an infinite series don't approach a limit, the series is called divergent.

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

Integral test.

The given series $\sum_{n = 3}^{\infty} \frac{1}{n^{2} - 4}$ .

Use the integral in the given series, $\int_{}^{\infty} \frac{1}{n^{2} - 4} d n$ .

The integral is also written as follows:

$\int_{}^{\infty} \frac{1}{n^{2} - 4} d n = \int_{}^{\infty} [\frac{1}{4 (n - 2)} - \frac{1}{4 (n + 2)}] d n$

Solve integral.

Solve the integral is as follows:

Use the integral formula, $\int \frac{1}{x} d x = \ln x + c$ , where is a constant.

$\begin{array}{rcl} \int_{}^{\infty} [\frac{1}{4 (n - 2)} - \frac{1}{4 (n + 2)}] d n & = & {[\frac{1}{4} \ln (n - 2) - \frac{1}{4} \ln (n + 2)]}^{\infty} \\ = & \frac{1}{4} \ln (\infty - 2) - \frac{1}{4} \ln (\infty + 2) \\ = & \infty \end{array}$

Hence, the series approaches to infinite therefore the given series diverges.

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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.
$\sum_{n = 3}^{\infty} \frac{1}{n^{2} - 4}$

Short Answer

Step by step solution

Definition of convergent and divergent.

Integral test.

Solve 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.∑n=3∞1n2-4

Short Answer

Step by step solution

Definition of convergent and divergent.

Integral test.

Solve integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Electrostatics

Astrophysics

Waves Physics

Medical Physics

Circular Motion and Gravitation

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.
$\sum_{n = 3}^{\infty} \frac{1}{n^{2} - 4}$