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

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Study anywhere. Anytime. Across all devices.

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