PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Sequences & Series (Convergence)
Let \(\rm a_n=\sin \left(\frac{1}{n^5}\right)\)and \(\rm b_n=\sin \left(\frac{1}{\sqrt{n}}\right)\) for n ∈ ℕ. Then
1
both \(\rm \displaystyle \sum_{n=1}^{\infty} a_n\) and \( \rm \displaystyle\sum_{n=1}^{\infty} b_n\) are convergent
2
\(\rm \displaystyle\sum_{n=1}^{\infty} a_n\) is convergent but \(\rm \displaystyle \sum_{n=1}^{\infty} b_n\) is NOT convergent
3
\(\rm \displaystyle \sum_{n=1}^{\infty} a_n\) is NOT convergent but \(\rm \displaystyle \sum_{n=1}^{\infty} b_n \) is convergent
4
both \(\rm \displaystyle \sum_{n=1}^{\infty} a_n\) and \(\rm \displaystyle \sum_{n=1}^{\infty} b_n \) are NOT convergent