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Chapter 7: Q. 38 (page 640)

In Exercises 35鈥�40 use the root test to analyze whether the given series converges or diverges. If the root test is inconclusive, use a different test to analyze the series.

鈭�k=2鈭�1lnkk

Short Answer

Expert verified

The given series diverges.

Step by step solution

01

Step 1. Given Information.

The given series is鈭�k=2鈭�1lnkk.

02

Step 2. Determine whether the given series converges or diverges. 

By using the root test, let the general term isak=1lnkk.

So,

蚁=limk鈫�鈭�ak1k蚁=limk鈫�鈭�1lnkk1k蚁=limk鈫�鈭�1lnk蚁=1

Since it is 1,thus the root test is inconclusive.

03

Step 3. Using the different test to analyze the series. 

We will use the divergence test to analyze the series. The divergence test states that if limk鈫�鈭�akdoes not exist or if limk鈫�鈭�ak鈮�0,then the series 鈭�k=1鈭�akis divergent.

So, let the sequence is ak=1lnkk.

Now, limk鈫�鈭�ak鈮�0.

Hence the series diverges.

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Most popular questions from this chapter

An Improper Integral and Infinite Series: Sketch the function f(x)=1xfor x 鈮� 1 together with the graph of the terms of the series 鈭�k=1鈭�1k.Argue that for every term Snof the sequence of partial sums for this series,Sn>鈭�1n+11xdx. What does this result tell you about the convergence of the series?

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