Chapter 2: Q16P (page 59)

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

Short Answer

Expert verified

The region of convergent is $|z - 3 + i| < \frac{1}{\sqrt{2}}$ .

Step by step solution

Given data

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

Concept of Ration test

The ratio test is a check (or "criterion") to see if a series will eventually converge to $\sum_{n - 1}^{\infty} a_{n}$ . Every term is a real or complex number, and when n is big, is not zero.

Calculation to check the series is convergent

Find $A_{n}$ and $A_{n + 1}$ as follows:

$A_{n} = 2^{n} {(z - 3 + i)}^{2 n} A_{n + 1} = 2^{n + 1} {(z - 3 + i)}^{2 n + 2}$

Apply ratio test as follows:

$ρ = \underset{n \to \infty}{l i m} |\frac{A_{n + 1}}{A_{n}}| ρ = \underset{n \to \infty}{l i m} |\frac{2^{n + 1} {(z - 3 + i)}^{2 n + 2}}{2^{n} {(z - 3 + i)}^{2 n}}| ρ = |2 {(z - 3 + i)}^{2}|$

If, $ρ < 1$ , then the series is convergence.

Calculation to find the region of convergent

The region of convergence is given as follows:

$|2 {(z - 3 + i)}^{2}| < 1 |{(z - 3 + i)}^{2}| < \frac{1}{2} |z - 3 + i| < \frac{1}{\sqrt{2}}$

Let, $z = x + i y$

Therefore, calculate further as follows:

$|x + i y - 3 + i| < \frac{1}{\sqrt{2}} |(x - 3) + (y + 1) i| < \frac{1}{\sqrt{2}} \sqrt{{(x - 3)}^{2} + {(y + 1)}^{2}} < \frac{1}{\sqrt{2}} {(x - 3)}^{2} + {(y + 1)}^{2} < \frac{1}{\sqrt{2}}$

It is an equation of disk with center $(3, - 1)$ and $r = \frac{1}{2}$ .

Hence, the region of convergent is $|z - 3 + i| < \frac{1}{\sqrt{2}}$ .

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

Short Answer

Step by step solution

Given data

Concept of Ration test

Calculation to check the series is convergent

Calculation to find the region of convergent

One App. One Place for Learning.

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Find the disk on convergence for each of the following complex power series.∑n=1∞2n(z+i-3)2n

Short Answer

Step by step solution

Given data

Concept of Ration test

Calculation to check the series is convergent

Calculation to find the region of convergent

One App. One Place for Learning.

Most popular questions from this chapter

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