GATE 2025 ES – Question 13
Classify $P=\sum_{n=1}^\infty1/n$ and $Q=\sum_{n=1}^\infty1/n^2$.
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Correct answer: (B) P divergent;Q convergent
Explanation
The p-series $\sum1/n^p$ converges precisely when p>1. P is the harmonic series with p=1 and diverges; Q has p=2 and converges. For example, the integral tests compare their tails to $\int_1^\infty dx/x$ and $\int_1^\infty dx/x^2$. Thus **B**.