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Chapter 7: Q. 16 (page 639)

Explain why the series 鈭�k=1鈭�1k3 converges. Which convergence tests could be used to prove this?

Short Answer

Expert verified

Hence proved.

Step by step solution

01

Step 1. Given information.

We are given鈭�k=1鈭�1k3.

02

Step 2. Explanation.

Now, According to Divergence and convergence test,

鈭�k=1鈭�1k3is of the form鈭�k=1鈭�1kp, wherep=3.p>1, therefore, the series鈭�k=1鈭�1k3converges.

The test which could be used to test the convergence of the series are Ratio test, Root Test.

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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.

2.2131313...

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series 鈭�k=1鈭�akin which ak鈫�0.

(b) A divergent p-series.

(c) A convergent p-series.

Find an example of a continuous function f :[1,鈭�)鈫�Rsuch that 鈭�1鈭�fxdxdiverges and localid="1649077247585" 鈭�k=1鈭�fkconverges.

Consider the series

Fill in the blanks and select the correct word:

Iflimk鈫�鈭�akbk=鈭�and鈭�k=1鈭�_____divergesthen鈭�k=1鈭�_____(converges/diverges).
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