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Chapter 7: Q. 4 (page 631)

Use the comparison test to explain why the series 鈭�k=1鈭�1k伪diverges when 伪is an integer greater than 1

Short Answer

Expert verified

The series鈭�k=1鈭�1k伪diverges when伪is greater than1

Step by step solution

01

Step 1. Given information

An series is given as鈭�k-1鈭�1k伪

02

Step 2. Applying comparison test

Terms of the given series are positive.

Now the series 鈭�k=1鈭�bkcan be written as

鈭�k=1鈭�bk=1ka

After that the ratio is limk鈫�鈭�akbkcan be written as:

limk鈫�鈭�akbk=limk鈫�鈭�1ka1ka=limi鈫�鈭�1=1

The value of limk鈫�鈭�akbk=1is a non-zero finite number.

Considering the conditions a>1&1a<1, the series 鈭�k=1鈭�bk=鈭�k=1鈭�1k1/ais divergent by p-series. Therefore the series鈭�k=1鈭�ak is also divergent

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