Question

For \( c \in \mathbb{R} \), let the sequence \( \{ u_n \} \) be defined by \[ u_n = \frac{(1 + \frac{c}{n})^{n^2}}{(3 - \frac{1}{n})^n} \] Then the values of \( c \) for which the series \[ \sum_{n=1}^{\infty} u_n \] converges are

• A. \( \log_e 6<c<\log_e 9 \)
• B. \( c<\log_e 3 \)
• C. \( \log_e 9<c<\log_e 12 \)
• D. \( \log_e 3<c<\log_e 6 \)

Hint:

For series convergence tests, the root and ratio tests are effective for sequences with exponential growth.

Answer 1

The Correct Option is B

Step 1: Understanding the sequence.
We need to determine the values of \( c \) for which the series \( \sum_{n=1}^{\infty} u_n \) converges. The sequence \( u_n \) involves powers and exponents that suggest exponential growth.
Step 2: Apply the root test.
Using the root test for convergence, we find the condition on \( c \) that ensures the series converges.
Step 3: Final conclusion.
The series converges for \( c<\log_e 3 \), and the correct answer is (B).