Chapter 2: Q6P (page 59)

Find the disk of convergence for each of the following complex power series.
$\sum_{n - 1}^{\infty} n^{2} {(3 i z)}^{n} .$

Short Answer

Expert verified

The region of convergence is $|z| < \frac{1}{3} . |$ .

Step by step solution

Given Information.

The given power series, i.e., $S_{n} = \sum_{n - 1}^{\infty} n^{2} {(3 i z)}^{n}$

Definition of Region of convergence.

The disc of convergence can be defined as the interior of the set of terms/points of the any converging series.

Find the Region of convergence.

Use the ratio test.

$ρ_{n} = \frac{a_{n + 1}}{a_{n}} ρ_{n} = \frac{{(n + 1)}^{2} {(3 iz)}^{n + 1}}{n^{2} {(3 iz)}^{n}}$

Calculate the value of $ρ$ , i.e.,

$ρ = \lim_{n \to \infty} |\frac{{(n + 1)}^{2} {(3 i z)}^{n + 1}}{n^{2} {(3 i z)}^{n}}| = \lim_{n \to \infty} |{(\frac{n + 1}{n})}^{2} (3 i z)| = \lim_{n \to \infty} |{(1 + \frac{1}{n})}^{2} (3 i z)| = |3 i z|$

$S_{n}$ is convergent for $ρ < 1$ , i.e., $|3 i z| < 1$ .

Rewrite $|3 i z| < 1$ as

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Find the disk of convergence for each of the following complex power series.
$\sum_{n - 1}^{\infty} n^{2} {(3 i z)}^{n} .$

Short Answer

Step by step solution

Given Information.

Definition of Region of convergence.

Find the Region of convergence.

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Find the disk of convergence for each of the following complex power series.∑n-1∞n2(3iz)n.

Short Answer

Step by step solution

Given Information.

Definition of Region of convergence.

Find the Region of 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.

Find the disk of convergence for each of the following complex power series.
$\sum_{n - 1}^{\infty} n^{2} {(3 i z)}^{n} .$