Question:medium
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:
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:
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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.
Updated On: Nov 28, 2025
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