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Chapter 7: Q. 2 (page 655)

Fill in the blanks.

For r_____ , the sequence rkconverges to _____.

Short Answer

Expert verified

The required answer is for r=1, the sequence rkconverges to 1 and for r<1, the sequence converges to 0.

Step by step solution

01

Step 1. Given Information 

The given data is that geometric sequence converges.

02

Step 2. Explanation 

Using the concept of convergence and divergence of Geometric series.

Let rkk=0鈭�be a geometric sequence. Then for r=1, the sequence converges to 1 and for r<1, the sequence converges to 0.

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Most popular questions from this chapter

Let 0 < p < 1. Evaluate the limitlimk鈫�鈭�1/klnk1/kp

Explain why we cannot use a p-series with 0 < p < 1 in a limit comparison test to verify the divergence of the series鈭�k=2鈭�1klogk

Determine whether the series 鈭�k=0鈭�-98kconverges or diverges. Give the sum of the convergent series.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

鈭�k=3鈭�鈥�1(k鈭�2)2

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If ak鈫�0, then 鈭�k=1鈭�akconverges.

(b) True or False: If 鈭�k=1鈭�akconverges, then ak鈫�0.

(c) True or False: The improper integral 鈭�1鈭�f(x)dxconverges if and only if the series 鈭�k=1鈭�f(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series 鈭�k=1鈭�k-pconverges.

(f) True or False: If f(x)鈫�0as x鈫�鈭�, then 鈭�k=1鈭�f(k) converges.

(g) True or False: If 鈭�k=1鈭�f(k)converges, then f(x)鈫�0as x鈫�鈭�.

(h) True or False: If 鈭�k=1鈭�ak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

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 thatRn鈮�10-6

鈭�k=1鈭�1k2
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