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Chapter 7: Q. 12 (page 652)

Explain why you must use two convergence tests to show that a series鈭�k=1鈭�akConvergesconditionally

Short Answer

Expert verified

Forconditionallyconvergentseries鈭�k=1鈭�akmustdivergeand鈭�k=1鈭�akconverges.

Step by step solution

01

Step 1. Given

鈭�k=1鈭�akConvergesconditionally.

02

Step 2. Explanation

Theseries鈭�k=1鈭�akwillbeconvergeconditionallyif鈭�k=1鈭�akDiverges.Hence,forconditionallyconvergentseries鈭�k=1鈭�akmustdivergeand鈭�k=1鈭�akconverges.Hence,forconditionallyconvergentseriestwotestarerequired.

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Most popular questions from this chapter

Given that a0=-3,a1=5,a2=-4,a3=2and 鈭�akk=2鈭�=7, find the value of鈭�akk=0鈭�.

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?

In Exercises 48鈥�51 find all values of p so that the series converges.

鈭�k=1鈭�lnkkp

Determine whether the series 鈭�n=1鈭�-1n+13nconverges or diverges. Give the sum of the convergent series.

Explain why, if n is an integer greater than 1, the series 鈭�k=1鈭�1kn diverges.
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