Limits
What is the value of \lim_{n \to \infty} \left( \frac{1}{n^2} + \frac{2}{n^2} + \cdots + \frac{n}{n^2} \right), a sum-to-limit problem from sequences?
Select the correct option:
Solution
21
The given expression is a finite arithmetic sum divided by n^2, and the cleanest method is to evaluate the sum in closed form before passing to the limit. The numerator 1 + 2 + \cdots + n equals n(n+1)/2, so the whole expression becomes \frac{n(n+1)}{2n^2} = \frac{n+1}{2n} = \frac{1}{2} + \frac{1}{2n}. As n \to \infty the second term vanishes, leaving the limit 1/2. Equivalently this is the Riemann sum for \int_0^1 x,dx = 1/2, reinforcing the answer geometrically. Option 0 treats each term as negligible individually but forgets that there are n terms whose total is of order n. Option 1/4 corresponds to \int_0^1 x,dx with a mistaken halving or to summing only half the terms. Option 1 ignores the division by 2 in the arithmetic series formula. The decisive JEE pattern is recognizing an averaged sum as a definite integral or closed arithmetic series. Plausibility check: for n = 1000 the value is 0.5005, hugging 1/2 from above exactly as the algebra predicts, which confirms the limit.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- limits
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
21
The given expression is a finite arithmetic sum divided by n^2, and the cleanest method is to evaluate the sum in closed form before passing to the limit. The numerator 1 + 2 + \cdots + n equals n(n+1)/2, so the whole expression becomes \frac{n(n+1)}{2n^2} = \frac{n+1}{2n} = \frac{1}{2} + \frac{1}{2n}. As n \to \infty the second term vanishes, leaving the limit 1/2. Equivalently this is the Riemann sum for \int_0^1 x,dx = 1/2, reinforcing the answer geometrically. Option 0 treats each term as negligible individually but forgets that there are n terms whose total is of order n. Option 1/4 corresponds to \int_0^1 x,dx with a mistaken halving or to summing only half the terms. Option 1 ignores the division by 2 in the arithmetic series formula. The decisive JEE pattern is recognizing an averaged sum as a definite integral or closed arithmetic series. Plausibility check: for n = 1000 the value is 0.5005, hugging 1/2 from above exactly as the algebra predicts, which confirms the limit.
This easy difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of limits. It appeared in the 2025 exam.
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