Consider the statements. I. if \(s_{n} = \frac{1}{n}\), the sequence \(\{s_{n}\}\) is divergent. II. if \(s_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\), the sequence \(\{s_{n}\}\) is convergent. Which one of the following is correct?
Show Hint
Do not confuse the convergence of a sequence $\{s_n\}$ with the convergence of a series $\sum s_n$.
The sequence $\{1/n\}$ converges to 0, but the harmonic series $\sum 1/n$ diverges to infinity.
Step 1: Understanding the Question:
The question requires us to evaluate the convergence behavior of two mathematical sequences. Step 2: Key Formula or Approach:
A sequence \(\{s_{n}\}\) is convergent if its limit as \(n \to \infty\) is a finite, real number:
\[ \lim_{n \to \infty} s_{n} = L \]
If the limit is infinite or does not exist, the sequence is divergent. Step 3: Detailed Explanation:
• Statement I analysis:
The sequence is defined as \(s_{n} = \frac{1}{n}\).
Taking the limit as \(n \to \infty\):
\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
Since the limit exists and is a finite number (0), the sequence is convergent. Therefore, Statement I is false.
• Statement II analysis:
The sequence is defined as \(s_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\).
This sequence represents the sequence of partial sums of the harmonic series \(\sum_{k=1}^{\infty} \frac{1}{k}\).
The harmonic series is a well-known divergent series.
Because the series diverges, the sequence of its partial sums grows without bound:
\[ \lim_{n \to \infty} s_{n} = \infty \]
Since the limit is infinite, the sequence \(\{s_{n}\}\) is divergent. Therefore, Statement II is false. Step 4: Final Answer:
Since both statements I and II are false, the correct option is "Neither I nor II is true".
