Chapter 2: Q3P (page 59)

Find the disk of convergence for each of the following complex power series.
$1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - \cdot \cdot \cdot$

Short Answer

Expert verified

The entire complex plane is the region of convergence.

Step by step solution

Given Information.

The given power series, i.e., $S_{n} = 1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - \cdot \cdot \cdot$

Definition of Disc of convergence.

Disc of convergence is defined as the interior of the set of points of convergence of a power series, whose radius is defined as the series' convergence radius.

Find the general term of the series.

Use the series to find the general term.

$S_{n} = 1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - \cdot \cdot \cdot \cdot \cdot \cdot (1) S_{n} = \underset{}{\sum \frac{{(- 1)}^{n} z^{2 n}}{(n + 1)}} \cdot \cdot \cdot (2)$

Find the Region of convergence.

Use the ratio test.

$ρ_{n} = \frac{a_{n + 1}}{a_{n}} ρ_{n} = |\frac{\frac{{(- 1)}^{n + 1} z^{2 (n + 1)}}{(n + 2)!}}{\frac{{(- 1)}^{n} z^{2 n}}{(n + 1)!}}| = |- \frac{z^{2}}{n + 2}|$

Calculate the value of $ρ$ , i.e.,

$\begin{array}{l} ρ = \lim_{n \to \infty} ρ_{n} \\ = \lim_{n \to \infty} |- \frac{z^{2}}{n + 2}| \\ = 0 \end{array}$

$S_{n}$ is convergent for all values of z.

Hence, the entire complex plane is the region of convergence.

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.

Find the disk of convergence for each of the following complex power series.
$1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - \cdot \cdot \cdot$

Short Answer

Step by step solution

Given Information.

Definition of Disc of convergence.

Find the general term of the series.

Find the Region of convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Fields in Physics

Kinematics Physics

Rotational Dynamics

Fundamentals of Physics

Radiation

Study anywhere. Anytime. Across all devices.

Find the disk of convergence for each of the following complex power series.1-z23!+z45!-···

Short Answer

Step by step solution

Given Information.

Definition of Disc of convergence.

Find the general term of the series.

Find the Region of convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Modern Physics

Fields in Physics

Kinematics Physics

Rotational Dynamics

Fundamentals of Physics

Radiation

Study anywhere. Anytime. Across all devices.

Find the disk of convergence for each of the following complex power series.
$1 - \frac{z^{2}}{3!} + \frac{z^{4}}{5!} - \cdot \cdot \cdot$