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91Ó°ÊÓ

In Exercises 35–40 use the root test to analyze whether the given series converges or diverges. If the root test is inconclusive, use a different test to analyze the series.

∑k=1∞5kk24k+1

Short Answer

Expert verified

The given series is diverges.

Step by step solution

01

Step 1. Given Information.

The given series is∑k=1∞5kk24k+1.

02

Step 2. Determine whether the given series converges or diverges. 

By using the root test, let the general term is ak=5kk24k+1.

So,

ÒÏ=limk→∞ak1kÒÏ=limk→∞5kk24k+11kÒÏ=limk→∞5k1kk21k4k+11kÒÏ=limk→∞5k1k24k+11kÒÏ=514ÒÏ=54

Since role="math" localid="1649146831183" 54>1,by using the root test the given series is diverges.

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