Determine whether the series converges or diverges. sum_{n=2}^inftyfrac{1}{nln n}

Zoe Oneal 2021-03-07 Answered
Determine whether the series converges or diverges.
n=21nlnn
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Expert Answer

Ayesha Gomez
Answered 2021-03-08 Author has 104 answers

Given that:
The series n=21nlnn
By using,
Limit - Comparison test :
Suppose that we have two series an and bn with an0,bn>0 for all n
Define c=limnanbn
If c is positive (that is c>0) and is finite then either both series converges or both series diverge.
P - series test:
The series is of the form 1np is converges if p>1 and diverges if 0<p1
Let ,
The original series is an=1nlnn
We need to find the series similar to the original series but simpler.
1nlnn<1n
Then,
The comparison series is bn=1n
By using the p- series test,
n=21n=n=21n1
Here p = 1
Then,
By the p - series test bn is divergent.
c=limnanbn
=limn1nlnn×n1
=limn1lnn
=diverge.
Then,
By the Limit - Comparison Test,
The series n=21nlnn is diverges.
Therefore,
The series n=21nlnn is diverges.

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