Chapter 2: Q6P (page 57)

Test each of the following series for convergence.
$\underset{}{\sum \frac{1 + i}{n^{2}}}$

Short Answer

Expert verified

The series is convergent.

Step by step solution

Given Information

The series is $\underset{}{\sum \frac{1 - i}{n^{2}}}$ .

Definition of the Convergent series.

A series can be categorized to be convergent if the terms of a series propagate to zero when the number of terms moves towards infinity.

Test the convergence.

The series is $\sum_{} \frac{1 + i}{n^{2}}$ .

$A_{n} = \frac{1 + i}{n^{2}} A_{n + 1} = \frac{1 + i}{{(n + 1)}^{2}}$

Find the limit $\lim_{n \to \infty} |\frac{A_{n + 1}}{A_{n}}|$ .

$p = \lim_{n \to \infty} |\frac{A_{n + 1}}{A_{n}}| = \lim_{n \to \infty} {|\frac{n}{n + 1}|}^{2} = \lim_{n \to \infty} {|\frac{1}{1 - \frac{1}{n}}|}^{2} = 1$

Test fails.

Find the sum of the series.

$S = \int_{0}^{\infty} (\frac{1 + i}{n^{2}}) dn = {[\frac{- (1 + i)}{n}]}_{1}^{\infty} = 1 + i S = \sqrt{2}$

The sum is finite.

Hence the series is convergent.

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Test each of the following series for convergence.
$\underset{}{\sum \frac{1 + i}{n^{2}}}$

Short Answer

Step by step solution

Given Information

Definition of the Convergent series.

Test the convergence.

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Test each of the following series for convergence.∑1+in2

Short Answer

Step by step solution

Given Information

Definition of the Convergent series.

Test the convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Engineering Physics

Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.
$\underset{}{\sum \frac{1 + i}{n^{2}}}$