PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Sequences & Series (Convergence)
For a, b, c \( \in \mathbb{R} \) with \( b \neq 0\) and n \(\in \mathbb{N}\) , let
\(a_n = a^n n^b \left( \frac{n + 3}{n + 1} \right)^{n^2} \quad \)and \( \quad c_n = \frac{n^n}{n!} \cdot \frac{1}{b^n} \cdot \sqrt[n]{\frac{n + 3}{n}}\).
Then, which one of the following statements is TRUE?
1
If |a| < e2 and b > 0 , then \(\sum_{n=1}^{\infty} a_n \) is convergent.
2
If |a| > e2 and b > 2 , then \(\sum_{n=1}^{\infty} a_n\) is convergent.
3
If 1 < b < e , then \( \sum_{n=1}^{\infty} c_n \) is convergent.
4
If |b| < e , then\( \sum_{n=1}^{\infty} c_n \) is convergent.