Question

Evaluate the limit \[ \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \dots + \frac{1}{n}}{(n + e^n)^{1/n} \log_e n} \]

• A. \( \frac{1}{\pi} \)
• B. \( \frac{1}{e} \)
• C. \( \frac{e}{\pi} \)
• D. \( \frac{\pi}{e} \)

Hint:

For large \( n \), the harmonic sum behaves like \( \log_e n \), and terms like \( e^n \) dominate in expressions with sums involving large \( n \).

Answer 1

The Correct Option is A

Step 1: Simplify the numerator.
The numerator is the harmonic sum: \[ \sum_{k=1}^{n} \frac{1}{k} \] As \( n \to \infty \), this sum behaves asymptotically as \( \log_e n \). Step 2: Simplify the denominator.
The denominator contains \( (n + e^n)^{1/n} \). Since \( e^n \) grows much faster than \( n \), we have: \[ (n + e^n)^{1/n} \sim e \] Thus, the denominator behaves like \( e \log_e n \) as \( n \to \infty \). Step 3: Final simplification.
Now, the limit becomes: \[ \frac{\log_e n}{e \log_e n} = \frac{1}{e} \] Thus, the final result is \( \frac{1}{e} \).