Question

For the sequence \(s_n = \frac{b^n}{n^2}\), which of the following is true?

• A. \(\lim s_n = 1\)
• B. \((s_n)\) diverges for \(0 < b < 1\)
• C. \((s_n)\) diverges for \(b > 1\)
• D. \((s_n)\) converges to 1 for \(0 < b < 1\)

Hint:

Exam Tip:
For sequences involving exponentials and polynomials:

• Exponential growth dominates polynomial growth.
• If \(b > 1\), the sequence diverges.
• If \(0 < b < 1\), the sequence converges to 0.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to analyze the convergence of the sequence \(s_n = \frac{b^n}{n^2}\) for different values of \(b\).

Step 2: Key Formula or Approach:

• If \(0 < b < 1\), then \(b^n \to 0\) and \(n^2 \to \infty\), so \(\frac{b^n}{n^2} \to 0\).
• If \(b = 1\), then \(s_n = \frac{1}{n^2} \to 0\).
• If \(b > 1\), then \(b^n\) grows exponentially while \(n^2\) grows polynomially, so \(s_n \to \infty\) (diverges).

Step 3: Analyzing the Options:

• (A) \(\lim s_n = 1\): False.
• (B) \((s_n)\) diverges for \(0 < b < 1\): False, it converges to 0.
• (C) \((s_n)\) diverges for \(b > 1\): True.
• (D) \((s_n)\) converges to 1 for \(0 < b < 1\): False, it converges to 0.

Step 4: Final Answer:
Therefore, option (C) is correct.