Question

Consider the following two series:
\[ \textbf{P: } \sum_{n=1}^{\infty} \frac{1}{n} \qquad \textbf{Q: } \sum_{n=1}^{\infty} \frac{1}{n^2} \]
Choose the correct option from the following:

• A. P is convergent series; Q is divergent series
• B. P is divergent series; Q is convergent series
• C. Both P and Q are convergent series
• D. Both P and Q are divergent series

Hint:

Remember: Harmonic series \( \sum \frac{1}{n} \) always diverges. Any p-series \( \sum \frac{1}{n^p} \) with \( p>1 \) converges.

Answer 1

The Correct Option is B

Step 1: Analyze Series \( P = \sum_{n=1}^{\infty} \frac{1}{n} \)
This is the harmonic series. It is a known result that the harmonic series: \[ \sum_{n=1}^{\infty} \frac{1}{n} \] is divergent even though the terms go to 0. So \( P \) is divergent.

Step 2: Analyze Series \( Q = \sum_{n=1}^{\infty} \frac{1}{n^2} \)
This is a \( p \)-series with \( p = 2 \). The rule is: \[ \sum \frac{1}{n^p} \text{ converges if } p > 1 \] Since \( p = 2 > 1 \), \( Q \) is convergent.