Chapter 1: Q12P (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.
$\sum_{1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}}$

Short Answer

Expert verified

The series $\sum_{1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}}$ 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 is $\sum_{1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}}$ .

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

Now solve the integral is as follows:

Let $t = n^{2} + 1$ , $d t = 2 n d n$

Solve integral.

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

$\int_{}^{\infty} \frac{d t}{2 t^{2}} = \int_{}^{\infty} \frac{1}{2} t^{- 2} d t$

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

$\begin{array}{rcl} \int_{}^{\infty} \frac{1}{2} t^{- 2} d t & = & \frac{1}{2} {[\frac{t^{- 2 + 1}}{- 2 + 1}]}^{\infty} \\ = & \frac{1}{2} {[\frac{t^{- 1}}{- 1}]}^{\infty} \\ = & - \frac{1}{2} [\infty^{- 1}] \\ = & \infty \end{array}$

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.
$\sum_{1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}}$

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.∑1∞n(n2+1)2

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

Rotational Dynamics

Waves Physics

Electrostatics

Astrophysics

Medical Physics

Electromagnetism

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.
$\sum_{1}^{\infty} \frac{n}{{(n^{2} + 1)}^{2}}$