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Lim n→∞(1^2+2^2+…+n^2/(n^3 is equal to Single correct option
lim n → ∞ 1 2 + 2 2 + ⋯ + n 2 n 3 \displaystyle\lim_{n\to\infty}\frac{1^2+2^2+\cdots+n^2}{n^3} n → ∞ lim n 3 1 2 + 2 2 + ⋯ + n 2 is equal to
A. 1 2 \dfrac12 2 1 B. 1 1 1 C. 1 6 \dfrac16 6 1 D. 1 3 \dfrac13 3 1 Show answer and solution Answer: (D) 1 3 \dfrac13 3 1
The sum is
n ( n + 1 ) ( 2 n + 1 ) 6 \dfrac{n(n+1)(2n+1)}{6} 6 n ( n + 1 ) ( 2 n + 1 ) , so the ratio is
( 1 + 1 / n ) ( 2 + 1 / n ) 6 → 2 6 = 1 3 \dfrac{(1+1/n)(2+1/n)}{6}\to\dfrac26=\dfrac13 6 ( 1 + 1/ n ) ( 2 + 1/ n ) → 6 2 = 3 1 . Equivalent to
∫ 0 1 x 2 d x \int_0^1x^2dx ∫ 0 1 x 2 d x . The value
1 / 6 1/6 1/6 is just the coefficient before taking the limit of the other factors.
Original question written in the NDA 2021 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake