Let's examine the convergence or divergence of series P and Q step by step:

• Analysis of Series P: \(\sum_{n=1}^{\infty} \frac{1}{n}\)
  • This series is the famous harmonic series.
  • The general term of the series is \(\frac{1}{n}\).
  • The harmonic series is known to be a divergent series.
  • Reason: The partial sum of the harmonic series grows without bound as we add more terms. Formally, it can be shown using the integral test or by comparison with the integral \(\int_{1}^{\infty} \frac{1}{x} \, dx\), which diverges.
• Analysis of Series Q: \(\sum_{n=1}^{\infty} \frac{1}{n^2}\)
  • This series is a p-series with \( p = 2 \).
  • The general term of the series is \(\frac{1}{n^2}\).
  • For a p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\), the series converges if \( p > 1 \).
  • Since \( p = 2 \) in this series, it converges.
  • Reason: The convergence of a p-series with \( p = 2 \) can be shown using the integral test or proven directly by comparing it to a converging integral, like \(\int_{1}^{\infty} \frac{1}{x^2} \, dx\), which converges.

Conclusion: Based on the explanations above, the correct option is that P is a divergent series; Q is a convergent series.