Q2P Find the disk of convergence for... [FREE SOLUTION]

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Chapter 2: Q2P (page 59)

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

Short Answer

Expert verified

The disc of convergence is $|z| < 1$ .

Step by step solution

Given Information.

The given power series, i.e., $S_{n} = z - \frac{Z^{2}}{2} + \frac{Z^{3}}{3} - \frac{z^{4}}{4} + 鈥�$

Definition of Disc of convergence.

The interior of the set of points of convergence of a power series is called the disc of convergence. Its radius is known as the series' convergence radius.

Find the general term of the series.

Use the series to find general terms.

$\begin{array}{l} S_{n} = z - \frac{z^{2}}{2} + \frac{z^{3}}{3} - \frac{z^{4}}{4} + . . . . . . (1) \end{array} S_{n} = \underset{}{{鈭�}_{n = 1}^{鈭�} \frac{{(- 1)}^{n + 1} z^{n}}{n}} . . . (2)$

Find the disc of convergence.

Use the ratio test.

$蚁_{n} = \frac{a_{n + 1}}{a_{n}} 蚁_{n} = |\frac{\frac{{(- 1)}^{n + 2} z^{n + 1}}{n + 1}}{\frac{{(- 1)}^{n + 1} z^{n}}{n}}| = |(- 1) z (\frac{n}{n + 1})| = |(- 1) z (\frac{1}{1 + 1 / n})|$

Calculate the value of $蚁$ , i.e.,

role="math" localid="1658731500708" $蚁 = \lim_{n 鈫� 鈭�} 蚁_{n} = \lim_{n 鈫� 鈭�} |(- 1) z (\frac{1}{1 + \frac{1}{n}})| = |z|$

$S_{n}$ is convergent for $蚁 < 1$ , i.e., $|z| < 1$ .

Hence, the disc of convergence is $|z| < 1$ .

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91影视

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

Short Answer

Step by step solution

Given Information.

Definition of Disc of convergence.

Find the general term of the series.

Find the disc of convergence.

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91影视

Find the disk of convergence for each of the following complex power series.z-z22+z33-z44+鈥�

Short Answer

Step by step solution

Given Information.

Definition of Disc of convergence.

Find the general term of the series.

Find the disc of convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Space Physics

Fundamentals of Physics

Particle Model of Matter

Measurements

Translational Dynamics

Radiation

Study anywhere. Anytime. Across all devices.

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