Question

\( \displaystyle \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \cdots + \frac{1}{n}}{n} \) is equal to :

• A. 0
• B. 1/2
• C. 1/e
• D. 1

Hint:

Cauchy's Theorem is a powerful tool to find the limit of the arithmetic mean of a sequence.

Answer 1

The Correct Option is A

Step 1: Let $a_n = 1/n$. We know $\lim_{n \to \infty} a_n = 0$.
Step 2: By Cauchy's First Theorem on Limits: If $\lim_{n \to \infty} a_n = L$, then $\lim_{n \to \infty} \frac{a_1 + a_2 + ....... + a_n}{n} = L$.
Step 3: Here $L = 0$, so the required limit is 0.