Chapter 2: Q11P (page 57)

Test each of the following series for convergence.
$\sum_{} {(\frac{2 + i}{3 - 4 i})}^{2 n}$

Short Answer

Expert verified

The series converges, i.e., $ρ < 1$ .

Step by step solution

Given Information.

The given series, i.e. $\sum_{} {(\frac{2 + i}{3 - 4 i})}^{2 n}$ ,

Definition of Convergent and Divergent series.

A convergent series is one in which the partial sums all gravitate to the same finite number, also known as a limit. Divergent refers to any series that is not converging.

Calculate the value of ρn.

Find the value of $a_{n}$ and $a_{n + 1}$

$a_{n} = {(\frac{2 + i}{3 - 4 i})}^{2 n} . . . (1) a_{n + 1} = {(\frac{2 + i}{3 - 4 i})}^{2 (n + 1)} a_{n + 1} = {(\frac{2 + i}{3 - 4 i})}^{2 n + 2} . . . (2)$

If $ρ < 1$ series converges, if $ρ > 1$ then diverges.

$ρ = \lim_{n \to \infty} ρ_{n}$

Test the series for convergence.

Use the ratio test.

$ρ_{n} = \frac{a_{n + 1}}{a_{n}} = {(\frac{2 + i}{3 - 4 i})}^{2 n + 2 - 2 n} = {(\frac{2 + i}{3 - 4 i})}^{2} = - 0.187 + 0.07 i$

Calculate the value of $ρ$ , i.e.,

$ρ = \lim_{n \to \infty} |ρ_{n}| = \lim_{n \to \infty} |- 0.187 + 0.07 i| ρ = 0.2$

Hence, the series converges, i.e., $ρ < 1$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Test each of the following series for convergence.
$\sum_{} {(\frac{2 + i}{3 - 4 i})}^{2 n}$

Short Answer

Step by step solution

Given Information.

Definition of Convergent and Divergent series.

Calculate the value of ρn.

Test the series for convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Medical Physics

Engineering Physics

Linear Momentum

Magnetism and Electromagnetic Induction

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.∑(2+i3-4i)2n

Short Answer

Step by step solution

Given Information.

Definition of Convergent and Divergent series.

Calculate the value of ρn.

Test the series for convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

Medical Physics

Engineering Physics

Linear Momentum

Magnetism and Electromagnetic Induction

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.
$\sum_{} {(\frac{2 + i}{3 - 4 i})}^{2 n}$