Question:
\( \displaystyle \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \cdots + \frac{1}{n}}{n} \) is equal to :
\( \displaystyle \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \cdots + \frac{1}{n}}{n} \) is equal to :
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Cauchy's Theorem is a powerful tool to find the limit of the arithmetic mean of a sequence.
Updated On: Feb 12, 2026
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The Correct Option is A
Solution and Explanation
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.
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.
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