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Find two divergent geometric series k=0akand k=0bkwith all positive terms such that k=0akbkconverges.

Short Answer

Expert verified

Ans:

part (a). The divergent geometric seriesk=0ak=k=02k

part (b). The divergent geometric seriesk=0bk=k=04k

part (c). The series is convergek=0akbk=k=012k

Step by step solution

01

Step 1. Given Information: 

Consider the two divergent geometric series k=0akand k=0bkwith all positive terms such that k=0akbkconverge.

02

Step 2. Finding the divergent geometric series  ∑k=0∞ak

Consider the geometric series k=0ak=k=02k.

The series k=0akis geometric series with common ratio r=2, which is greater than 1 . The geometric series with ratio greater than 1 is divergent.

Therefore, k=0ak=k=02kis divergent.

03

Step 3. Finding the divergent geometric series  ∑k=0∞bk

Consider the geometric series k=0bk=k=04k.

The series k=04kis geometric series with common ratio r=4, which is greater than 1 . The geometric series with ratio greater than 1 is divergent.

Therefore, k=0bk=k=04kis dlvergent.

04

Step 4. Finding the converges geometric series of ∑k=0 ∞ak bk with all positive terms:

The series k=0akbk is

k=0akbk=k=02k4k=k=012k

The series k=012xis a geometric series with common ratio r=12, which is less than 1 . The geometric series with ratio less than 1 is convergent.

Therefore, k=0akbk=k=012k is convergent.

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