Limits, Continuity & Differentiability
Differential Calculus-1
star_batch_jee_advanced_2025
Grade 12
Question:
Let $H_n$ denotes the harmonic mean of $n$ positive integers $n+1, n+2, n+3, \ldots, n+n$. If $\lim_{n \to \infty} \left( \frac{H_n}{n} \right) = \frac{1}{k}$ then the value of $e^k$ is ______.
Step-by-Step Solution
Key Concept: Recognizing a harmonic sum as a Riemann sum to evaluate its asymptotic behavior using integration.
Given $H_n = \frac{1}{n+1} + \frac{1}{n+2} + ... + \frac{1}{n+n}$, we find $\frac{n}{H_n} = \lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n+r}$ which can be rewritten as a Riemann sum: $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{1}{1+r/n} = \int_0^1 \frac{dx}{1+x} = \ln 2$. Therefore $\lim_{n \to \infty} \frac{H_n}{n} = \frac{1}{\ln 2}$.
Correct Answer: Looking at this problem, I need to find the limit and then calculate $e^k$.
From the solution:
- The harmonic mean of the $n$ integers is $H_n = \frac{n}{\sum_{r=1}^{n} \frac{1}{n+r}}$
- Using Riemann sum interpretation: $