PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Sequences & Series (Convergence)
Define the sequences \(\left\{c_n\right\}_{n=3}^{\infty}\) and \(\left\{d_n\right\}_{n=3}^{\infty}\) as
\(c_n\) = (log n +( log log n)2)log n and \(d_n\) = \(n^{\left(1+\frac{2}{\log n}\right)}\)
Which one of the following is TRUE?
1
\(\sum_{n=3}^{\infty} \frac{1}{c_n}\) is convergent but \(\sum_{n=3}^{\infty} \frac{1}{d_n}\) is divergent
2
\(\sum_{n=3}^{\infty} \frac{1}{c_n}\) is divergent but \(\sum_{n=3}^{\infty} \frac{1}{d_n}\) is convergent
3
Both \(\sum_{n=3}^{\infty} \frac{1}{c_n}\) and \(\sum_{n=3}^{\infty} \frac{1}{d_n}\) are divergent
4
Both \(\sum_{n=3}^{\infty} \frac{1}{c_n}\) and \(\sum_{n=3}^{\infty} \frac{1}{d_n}\) are convergent