GATE 2025 EC – Question 12
Consider the following series:
(i) $\sum_{n=1}^{\infty}\dfrac{1}{\sqrt n}$
(ii) $\sum_{n=1}^{\infty}\dfrac{1}{n(n+1)}$
(iii) $\sum_{n=1}^{\infty}\dfrac{1}{n!}$
Choose the correct option.
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Show answer and explanation
Correct answer: (B) Only (ii) and (iii) converge
Explanation
(i) is a p-series with $p=1/2\le1$, so it diverges. (ii) telescopes: $\frac1{n(n+1)}=\frac1n-\frac1{n+1}$, so the sum is 1. (iii) has sum $e-1$ (ratio test gives limit 0). So only (ii) and (iii) converge.