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Chapter 2: Q16P (page 59)

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

Short Answer

Expert verified

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

Step by step solution

01

Given data

The given series is, ${鈭�}_{n = 1}^{鈭�} 2^{n} {(z + i - 3)}^{2 n}$ .

02

Concept of Ration test

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

03

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:

$蚁 = \underset{n 鈫� 鈭�}{l i m} |\frac{A_{n + 1}}{A_{n}}| 蚁 = \underset{n 鈫� 鈭�}{l i m} |\frac{2^{n + 1} {(z - 3 + i)}^{2 n + 2}}{2^{n} {(z - 3 + i)}^{2 n}}| 蚁 = |2 {(z - 3 + i)}^{2}|$

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

04

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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Most popular questions from this chapter

Express the following complex numbers in the $x + i y$ form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others鈥攖ry it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

7. $3 e^{2 (1 + i 蟺)}$

Describe geometrically the set of points in the complex plane satisfying the following equations. .

Find each of the following in rectangular $a + {}_{z}b$ form if $z = 2 - {}_{t}3$ ; if $z = x + {}_{z}y$ .

$\frac{z}{z}$ .

Express the following complex numbers in the form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others鈥攖ry it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

19. ${(1 鈭� i)}^{8}$

Express the following complex numbers in the x+iy form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others鈥攖ry it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

23. $\frac{{(1 + i)}^{48}}{{(\sqrt{3} 鈭� i)}^{25}}$

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