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Chapter 2: Q12P (page 57)

Test each of the following series for convergence.

鈭�(3+2i)nn!

Short Answer

Expert verified

The series converges, i.e., 蚁<1.

Step by step solution

01

Given Information.

The given series, i.e.,鈭�(3+2i)nn!

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 anand an+1

role="math" localid="1658730311852" an=3+2inn!...(1)an=3+2in+1n+1!...(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 and put n+1!=n+1n!

蚁n=an+1an=3+2in+1(n+1)!脳n!3+2in蚁n=3+2in+13+2in脳n!(n+1)n!=3+2in+1

Calculate the value of 蚁, i.e.,

蚁=limn鈫�鈭�3+2in+1=3+2i鈭�=0

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

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Most popular questions from this chapter

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

(1+i1-i)5

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

|z+3i|=4.

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

|z-1|=1.

Question:First simplify each of the following numbers to the x+iyform or to there颈胃form. Then plot the number in the complex plane.

(3+i2+i)

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

3ii-3.
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