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Chapter 2: Q11P (page 57)

Test each of the following series for convergence.

鈭�(2+i3-4i)2n

Short Answer

Expert verified

The series converges, i.e., 蚁<1.

Step by step solution

01

Given Information.

The given series, i.e. 鈭�(2+i3-4i)2n,

02

Definition of Convergent and Divergent series.

A convergent series is one in which the partial sums all gravitate to the same finite number, also known as a limit. Divergent refers to any series that is not converging.

03

Calculate the value of  ρn.

Find the value of anandan+1

an=2+i3-4i2n...(1)an+1=2+i3-4i2(n+1)an+1=2+i3-4i2n+2...(2)

If 蚁<1series converges, if 蚁>1then diverges.

蚁=limn鈫�鈭�蚁n

04

Test the series for convergence.

Use the ratio test.

蚁n=an+1an=2+i3-4i2n+2-2n=2+i3-4i2=-0.187+0.07i

Calculate the value of 蚁, i.e.,

蚁=limn鈫�鈭�蚁n=limn鈫�鈭�-0.187+0.07i蚁=0.2

Hence, the series converges, i.e., 蚁<1.

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Most popular questions from this chapter

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly findin your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

5(cos0-isin0).

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

|e颈蟺1+i|

Test each of the following series for convergence.

鈭�(1+i)n

Test each of the following series for convergence.

鈭�1+in2

Describe geometrically the set of points in the complex plane satisfying the following equations. |z-1+i|=2.

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