Question

Define the sequences \(\left\{a_n\right\}^{\infin}_{n=3}\) and \(\left\{b_n\right\}^{\infin}_{n=3}\) as
\(a_n=(\log n+\log\ \log n)^{\log n}\) and \(b_n=n^{(1+\frac{1}{\log n})}\).
Which one of the following is TRUE ?

• A. \(\sum\limits^{\infin}_{n=3}\frac{1}{a_n}\) is convergent but \(\sum\limits_{n=3}^{\infin}\frac{1}{b_n}\) is divergent
• B. \(\sum\limits^{\infin}_{n=3}\frac{1}{a_n}\) is divergent but \(\sum\limits_{n=3}^{\infin}\frac{1}{b_n}\) is convergent
• C. Both \(\sum\limits_{n=3}^{\infin}\frac{1}{a_n}\) and \(\sum\limits_{n=3}^{\infin}\frac{1}{b_n}\) are divergent
• D. Both \(\sum\limits_{n=3}^{\infin}\frac{1}{a_n}\) and \(\sum\limits_{n=3}^{\infin}\frac{1}{b_n}\) are convergent

Answer 1

The Correct Option is A

The correct option is (A) : \(\sum\limits^{\infin}_{n=3}\frac{1}{a_n}\) is convergent but \(\sum\limits_{n=3}^{\infin}\frac{1}{b_n}\) is divergent.