GATE 2023 ES – Question 13
Classify the infinite series $P=\sum_{n=1}^\infty(n^2+1)/n^2$ and $Q=\sum_{n=1}^\infty(1+1/n)^{-n}$.
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Correct answer: (D) Both diverge
Explanation
P’s term is $1+1/n^2$,which tends to 1 rather than 0. Q’s term tends to $e^{-1}$,also nonzero. A necessary condition for series convergence is that its terms tend to 0; both fail it. Hence **both divergent, D**.