Chapter 2: Q10P (page 57)

Test each of the following series for convergence.
$\underset{}{\sum {(\frac{1 + i}{1 - i \sqrt{3}})}^{n}}$

Short Answer

Expert verified

The series is convergent.

Step by step solution

Given Information

The series is $\sum_{} {(\frac{1 + i}{1 - i \sqrt{3}})}^{n}$ .

Definition of the Convergent series.

A series is said to be convergent if the terms of a series get close to zero when the number of terms moves towards infinity.

Test the convergence.

The series is $\sum_{} {(\frac{1 + i}{1 - i \sqrt{3}})}^{n} .$ .

$A_{n} = \sum_{} {(\frac{1 + i}{1 - i \sqrt{3}})}^{n} A_{n + 1} = \sum_{} {(\frac{1 + i}{1 - i \sqrt{3}})}^{n + 1}$

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

$p = \lim_{n \to \infty} |\frac{A_{n + 1}}{A_{n}}| = \lim_{n \to \infty} |{(\frac{1 + i}{1 - i \sqrt{3}})}^{n + 1} \times {(\frac{1 - i \sqrt{3}}{1 + i})}^{n}| = \lim_{n \to \infty} |(\frac{1 + i}{1 - i \sqrt{3}})|$

p < 1, Hence the series is convergent.

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Test each of the following series for convergence.
$\underset{}{\sum {(\frac{1 + i}{1 - i \sqrt{3}})}^{n}}$

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+i1-i3)n

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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Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.
$\underset{}{\sum {(\frac{1 + i}{1 - i \sqrt{3}})}^{n}}$