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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=01k3+1

Short Answer

Expert verified

The given series converges.

Step by step solution

01

Step 1. Given Information.

The given series isk=01k3+1.

02

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

By using the root test, let the general term isak=1k3+1.

So,

ρ=limkak1kρ=limk1k3+11kρ=limk1k3+11kρ=limk1limkk3+11k

Let's solve denominator first,

limkk3+11kUseax=elnaxSo,k3+11k=elnk3+11kk3+11k=e1klnk3+1limkk3+11k=limke1klnk3+1limkk3+11k=elimk1klnk3+1limkk3+11k=e0limkk3+11k=1

Now,

ρ=limk1limkk3+11kρ=1

Since it is1,thus the root test is inconclusive.

03

Step 3. Using the different test to analyze the series.

We will use the comparison test to analyze the series. The comparison test states that let k=1akandk=1bkbe two series with positive terms such that 0akbk for every positive number k.If the series k=1bkthen the series k=1akalso converges.

Now, let the series k=0ak=k=01k3+1andk=0bk=k=01k3.

So, 01k3+11k3.

As we can see k=0bk=k=01k3is of the form k=0bk=k=01kp.

The p-series say that if p > 1 then it converges and if p < 1 then it diverges.

Here, p=3>1thus, the series converges.

If k=0bk=k=01k3convergesthank=0ak=k=01k3+1alsoconverges.

Hence, the given series converges.

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