Question

The value of \( \lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^{2}} \) is:

• A. 1
• B. 1/2
• C. 0
• D. 2

Hint:

For limits of rational functions at infinity, only the highest power terms in numerator and denominator matter.

Answer 1

The Correct Option is B

Step 1: Concept
Use the sum of the first $n$ natural numbers: $S_{n} = \frac{n(n+1)}{2}$.
Step 2: Analysis
$\lim_{n \to \infty} \frac{n(n+1)/2}{n^{2}} = \lim_{n \to \infty} \frac{n^{2} + n}{2n^{2}}$.
Step 3: Conclusion
Dividing by $n^{2}$: $\lim_{n \to \infty} \frac{1 + 1/n}{2} = 1/2$.
Final Answer: (B)