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Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

k=21klnk

Short Answer

Expert verified

Ans: The seriesk=21klnkis divergent.

Step by step solution

01

Step 1. Given information.

given,

k=21klnk

02

Step 2. The objective is to explain why the integral test is used to determine the convergence or divergence of the series and use the test to determine the convergence or divergence of the series. 

Consider function f(x)=1xlnx.

The function f(x)=1xlnxis continuous, decreasing, with positive terms. Therefore, all the conditions of the integral test are fulfilled. So, the integral test is applicable.

03

Step 3. Consider the integral ∫x=2∞ f(x)dx=∫x=2∞ 1xln⁡xdx

Therefore,

x=2f(x)dx=limkx=2k1xlnxdx=limku=ln2lnk1udu(Putlnx=u,1xdx=du)=limk[2u]ln2lnk=2limk[lnkln2](Substitution)=2(ln2)(Take limit)=

04

Step 4. Thus, the value of the integral is ∫x=2∞ 1xln⁡xdx=∞

The integral converges. Therefore, the series k=21klnkis divergent.

Hence, by integral test, the series k=21klnkis divergent

.

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