Question

Let \( a_n \) denote the term independent of \( x \) in the expansion of \( \left[ x + \frac{\sin(1/n)}{x^2} \right]^{3n} \). Then \( \lim_{n \to \infty} \frac{(a_n) n!}{3^n n^n} \) equals:

• A. \( 0 \)
• B. \( 1 \)
• C. \( e \)
• D. \( \frac{e}{\sqrt{3}} \)

Hint:

When evaluating the limit of terms from binomial expansions, carefully analyze the behavior of the terms as \( n \) grows large, especially when factorial and exponential terms are involved.

Answer 1

The Correct Option is A

We are tasked with finding the term independent of \( x \) in the expansion of: \[ \left[ x + \frac{\sin(1/n)}{x^2} \right]^{3n}. \] This involves using the binomial expansion and focusing on the term that is independent of \( x \).
Step 1: Approximate \( \sin(1/n) \) for large \( n \).
Since \( \sin(1/n) \approx \frac{1}{n} \) for large \( n \), we approximate: \[ \frac{\sin(1/n)}{x^2} \approx \frac{1}{n x^2}. \] Thus, the expression becomes: \[ \left[ x + \frac{1}{n x^2} \right]^{3n}. \]
Step 2: Binomial Expansion.
The binomial expansion of \( \left( x + \frac{1}{n x^2} \right)^{3n} \) is given by: \[ \left( x + \frac{1}{n x^2} \right)^{3n} = \sum_{k=0}^{3n} \binom{3n}{k} x^{3n-k} \left( \frac{1}{n x^2} \right)^k. \] Simplifying each term: \[ = \sum_{k=0}^{3n} \binom{3n}{k} \frac{x^{3n-k}}{n^k x^{2k}} = \sum_{k=0}^{3n} \binom{3n}{k} \frac{1}{n^k} x^{3n-3k-2k}. \] This simplifies to: \[ = \sum_{k=0}^{3n} \binom{3n}{k} \frac{1}{n^k} x^{3n-5k}. \]
Step 3: Finding the Independent Term.
For the term to be independent of \( x \), the exponent of \( x \) must be zero: \[ 3n - 5k = 0 \quad \Rightarrow \quad k = \frac{3n}{5}. \] Since \( k \) must be an integer, \( n \) must be a multiple of 5 for the term independent of \( x \) to exist.
Step 4: Asymptotic Behavior.
As \( n \) becomes very large, the coefficient of the term independent of \( x \) behaves in such a way that: \[ \lim_{n \to \infty} \frac{a_n n!}{3^n n^n} = 0. \] This is because the factorial growth of \( n! \) is not sufficient to offset the exponential decay due to the powers of \( n^n \) and \( 3^n \) in the denominator.
Step 5: Conclusion.
The value of the limit is \( 0 \).