PG entrance exam IIT JAM 2025 Mock Test Mathematical Science Analysis Sequences & Series (Convergence)
Let R1 and R2 be the radii of convergence of the power series \(\rm \displaystyle \sum_{n=1}^{\infty}(-1)^n x^{n-1} \) and \(\rm \displaystyle \sum_{n=1}^{\infty}(-1)^n \frac{x^{n+1}}{n(n+1)} \), respectively. Then
1
R1 = R2
2
R2 > 1
3
\(\rm \displaystyle \sum_{n=1}^{\infty}(-1)^n x^{n-1} \) converges for all x ∈ [−1, 1]
4
\(\rm \displaystyle\sum_{n=1}^{\infty}(-1)^n \frac{x^{n+1}}{n(n+1)}\) converges for all x ∈ [−1, 1]