Question

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

• A. 1
• B. \(\frac{1}{2}\)
• C. 0
• D. 2

Hint:

For limits at infinity of rational functions where the degree of the numerator equals the degree of the denominator:
The limit is simply the ratio of the coefficients of the highest power terms.
Here, the highest power is \( n^2 \), and the ratio of their coefficients is \( 1 / 2 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires finding the limit of a rational algebraic expression containing a sum in the numerator as the variable \( n \) approaches infinity.
Key Formula or Approach:
First, we simplify the sum in the numerator using the standard arithmetic progression formula for the sum of the first \( n \) natural numbers:
\[ \sum_{i=1}^n i = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2} \] Then, we substitute this formula back into the limit and evaluate it.

Step 2: Detailed Explanation:
Let us perform the mathematical steps:
-

Step 1: Substitute the sum formula into the limit expression:
Let the limit be denoted by \( L \).
\[ L = \lim_{n \to \infty} \frac{1 + 2 + 3 + \dots + n}{n^2} \] Using the sum formula:
\[ L = \lim_{n \to \infty} \frac{\frac{n(n+1)}{2}}{n^2} \] Simplify the fraction:
\[ L = \lim_{n \to \infty} \frac{n(n+1)}{2n^2} \] \[ L = \lim_{n \to \infty} \frac{n^2 + n}{2n^2} \] - Evaluate the limit as \( n \to \infty \):
To evaluate limits at infinity for rational functions, we divide both the numerator and the denominator by the highest power of \( n \) present in the denominator (which is \( n^2 \)):
\[ L = \lim_{n \to \infty} \frac{\frac{n^2}{n^2} + \frac{n}{n^2}}{\frac{2n^2}{n^2}} \] \[ L = \lim_{n \to \infty} \frac{1 + \frac{1}{n}}{2} \] Since \( \lim_{n \to \infty} \frac{1}{n} = 0 \), we substitute this value:
\[ L = \frac{1 + 0}{2} = \frac{1}{2} \]

Step 3: Final Answer:
The value of the limit is \( \frac{1}{2} \).
Therefore, the correct choice is Option (B).