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Chapter 7: Q. 22 (page 657)

For the series that follow, determine whether the series converges or diverges. Explain the criteria you are using and why your conclusion is valid.

∑k=1∞ 1k2k

Short Answer

Expert verified

The series converges.

Using the root test we can say that the series has positive terms. Then apply limits and find the solution.

Step by step solution

01

Part (a) Step 1. Given information.

Consider the given question,

∑k=1∞ 1k2k

02

Step 2. Use the root test.

According to the Root test,

∑k=1∞akis the series, if L=limk→∞ ak1/kthen,

First is if role="math" localid="1650691303843" L<1series converges.

Second is if L>1series diverges.

Third is if L=1the test is inconclusive.

The general term of the series ∑k=1∞ 1k2kis ak=1k2k.

Since, the series has positive terms, hence it meets the hypothesis of the Root test.

03

Step 3. Use the limit.

The value of limk→∞ ak1/kis given below,

limk→∞ ak1/k=limk→∞ 1k2k1k=limk→∞ 1k2k21k=limk→∞ 1k212=limk→∞ 1k=0

As the value of the limit is 1.

Therefore, the series ∑k=1∞ 1k2kconverges.

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