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Chapter 7: Q 68. (page 615)

Find the values of x for which the series 鈭�k=0鈭�3xkconverges.

Short Answer

Expert verified

The series 鈭�k=0鈭�3xk converges only for x>3or x<-3.

Step by step solution

01

Step 1. Given information.

Given a series 鈭�k=0鈭�3xk.

02

Step 2. Find all values of x for which the series converges.

A geometric series is of the form 鈭�k=0鈭�crk for some constants c and r.

Supposer is a non-zero real number, then 鈭�k=0鈭�crkconverges to c1-rif and only if r<1.

Here, the series 鈭�k=0鈭�3xkhas r=3x
.

For the series to converge, 3x<1.

Note that ify<a, then-a<y<a.

It follows that localid="1648831728426" 鈭�k=0鈭�3xkconverges for all localid="1648831751828" x>3or localid="1648831755416" x<-3.

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Most popular questions from this chapter

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that.

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Determine whether the series 9940-9920+9910-995+鈥�converges or diverges. Give the sum of the convergent series.

For a convergent series satisfying the conditions of the integral test, why is every remainder Rnpositive? How can Rnbe used along with the term Sn from the sequence of partial sums to understand the quality of the approximation Sn?

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

鈭�k=1鈭�鈥�12k鈭�3/4

Given thata0=-3,a1=5,a2=-4,a3=2and鈭�akk=2鈭�=7, find the value of鈭�akk=4鈭�.
See all solutions

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