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Chapter 7: Q. 18 (page 656)

Convergence or divergence of a series: For each of the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid.

鈭�k=1鈭�ln1k2

Short Answer

Expert verified

By divergence test, the given sequence diverges.

Step by step solution

01

Step 1. Given Information

The given sequence is鈭�k=1鈭�ln1k2.

02

Step 2. Apply Divergent test

  • The divergence test states that if the sequence akdoes not converge to 0, the series 鈭�k=1鈭�akdiverges.
  • In the given sequence, ak=ln1k2.
  • Find the value of limk鈫�鈭�ak.

limk鈫�鈭�ak=limk鈫�鈭�ln1k2=ln1鈭�2=ln0=-鈭�

  • Thus, by divergence test, the given sequence diverges.

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Most popular questions from this chapter

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

0.237237237...

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=0鈭�鈥�e鈭�k鈭�k=0鈭�鈥�e鈭�k

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

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=1鈭�鈥�(3k鈭�1)鈭�2/3
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