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Real Analysis Question on Sequences and Series

Which of the following statements is/are TRUE ?

A

∑n=1∞nlog⁡(1+1n3)\sum\limits_{n=1}^{\infin}n\log(1+\frac{1}{n^3}) is convergent

B

∑n=1∞(1−cos⁡(1n))log⁡n\sum\limits_{n=1}^{\infin}(1-\cos(\frac{1}{n}))\log n is convergent

C

∑n=1∞n2log⁡(1+1n3)\sum\limits_{n=1}^{\infin}n^2 \log(1+\frac{1}{n^3}) is convergent

D

∑n=1∞(1−cos⁡(1n))log⁡n\sum\limits_{n=1}^{\infin}(1-\cos(\frac{1}{\sqrt n}))\log n is convergent

Answer

∑n=1∞nlog⁡(1+1n3)\sum\limits_{n=1}^{\infin}n\log(1+\frac{1}{n^3}) is convergent

Explanation

Solution

The correct option is (A) : ∑n=1∞nlog⁡(1+1n3)\sum\limits_{n=1}^{\infin}n\log(1+\frac{1}{n^3}) is convergent and (B) : ∑n=1∞(1−cos⁡(1n))log⁡n\sum\limits_{n=1}^{\infin}(1-\cos(\frac{1}{n}))\log n is convergent.