Let [·] denote the greatest integer function and f(x)= _ n arrow ∞ 1 n^ 3 ∑_ k=1 ^ n [ k^ 2 3^ x ] . Then 12 ∑_ j=1 ^ ∞ f(j) is equal to \_\_\_\_ .

Correct answer: 0 Let [·] denote the greatest integer function. We have f(x) = _ n → ∞ (1)/(n^3) ∑_ k=1 ^n [(k^2)/(3^x)] . For large n : ∑_ k=1 ^n [(k^2)/(3^x)] ∑_ k=1 ^n (k^2)/(3^x) = (1)/(3^x) · (n(n+1)(2n+1))/(6) (n^3)/(3 · 3^x) Therefore: f(x) = _ n → ∞ (1)/(n^3) · (n^3)/(3 · 3^x) = 1 3^ x+1 Computing the infinite series: ∑_ j=1 ^∞ f(j) = ∑_ j=1 ^∞ 1 3^ j+1 = (1)/(3^2) + (1)/(3^3) + (1)/(3^4) + = (1)/(9) · (1 + (1)/(3) + (1)/(9) + ) = (1)/(9) · (1)/(1 - 1/3) = (1)/(9) · (3)/(2) = (1)/(6) Therefore: 12 ∑_ j=1 ^∞ f(j) = 12 · (1)/(6) = 2

JEE Main 202621 JanuaryEvening Shift

Exam: JEE Main
Year: 2026
Date: 21 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Limits

JEE Main 2026 · 21 January · Evening Shift · Mathematics · Limits

Let [·] denote the greatest integer function and f(x)= _ n arrow ∞...

Let

[\cdot]

denote the greatest integer function and

f(x)=\lim _{\mathrm{n} \rightarrow \infty} \frac{1}{\mathrm{n}^{3}} \sum_{\mathrm{k}=1}^{\mathrm{n}}\left[\frac{\mathrm{k}^{2}}{3^{x}}\right]

. Then

12 \sum_{\mathrm{j}=1}^{\infty} f(\mathrm{j})

is equal to

\_\_\_\_

Correct answer

Step-by-step solution

Let [·] denote the greatest integer function. We have

f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^n [\frac{k^2}{3^x}]

.
For large

n

\sum_{k=1}^n [\frac{k^2}{3^x}] \approx \sum_{k=1}^n \frac{k^2}{3^x} = \frac{1}{3^x} \cdot \frac{n(n+1)(2n+1)}{6} \approx \frac{n^3}{3 \cdot 3^x}

Therefore:

f(x) = \lim_{n \to \infty} \frac{1}{n^3} \cdot \frac{n^3}{3 \cdot 3^x} = \frac{1}{3^{x+1}}

Computing the infinite series:

\sum_{j=1}^\infty f(j) = \sum_{j=1}^\infty \frac{1}{3^{j+1}} = \frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4} + \cdots

= \frac{1}{9} \cdot (1 + \frac{1}{3} + \frac{1}{9} + \cdots) = \frac{1}{9} \cdot \frac{1}{1 - 1/3} = \frac{1}{9} \cdot \frac{3}{2} = \frac{1}{6}

Therefore:

12 \sum_{j=1}^\infty f(j) = 12 \cdot \frac{1}{6} = 2

Practice Limits on Quantrex Academy →

More from Limits

Let _ x → 2 (( (x-2))(rx^2 + (p-2)x - 2p))/((x-2)^2) = 5 for some r, p ∈ ℝ . If the set… JEE Main 2026 The value of _ x → 0 ((x^2 ^2 x)/(x^2 - ^2 x)) is: JEE Main 2026 Let f(x) = _ y → 0 ((1 - (xy)) (xy))/(y^3) . Then the number of solutions of the equation… JEE Main 2026 The product of all possible values of α , for which _ x → 0 ((1 - (α x) ((α+1)x)… JEE Main 2026 If _ x → 2 (x^3 - 5x^2 + ax + b) ( x-1 - 1) _e(x-1) = m , then a + b + m is equal to : JEE Main 2026 The value of _ x arrow 0 _ e ( (e x) · (e^ 2 x) · … · (e^ 10 x)) e^ 2 -e^ 2 x is equal to JEE Main 2026 If _ x arrow 0 e^ (a-1) x +2 ~b x+(c-2) e^ -x x x- _ e (1+x) =2 , then a^ 2 +b^ 2 +c^ 2… JEE Main 2026 Let f: R arrow(0, ∞) be a twice differentiable function such that f(3)=18, f^ (3)=0 and… JEE Main 2026 Let α and β be the real numbers such that _ x arrow 0 (1)/(x^3)((α)/(2) ∫_0^x (1)/(1-t^2)… JEE Advanced 2025 Let x_0 be the real number such that e^ x_0 +x_0=0 . For a given real number α , define… JEE Advanced 2025 Given below are two statements : Statement I : _ x arrow 0 ( ^ -1 x+ _e (1+x)/(1-x) -2 x… JEE Main 2025 If the function f(x)=( ( x)- ( x))/( x- x) is continuous at x=0 , then f(0) is equal to… JEE Main 2025 _ x arrow 0^ + (5(x)^ (1)/(3) ) _e(1+3 x^2) ( ^ -1 3 x )^2(e^ 5(x)^ (4)/(3) -1) is equal… JEE Main 2025 If _ x arrow 1^ + ((x-1)(6+λ (x-1))+μ (1-x))/((x-1)^3)=-1 , where λ, μ ∈ ℝ , then λ+μ is… JEE Main 2025 If Lim_ x arrow 0 (( x)/(x))^ (1)/(x^2) =p , then 96 _e p is equal to ______ JEE Main 2025 If _ x arrow 0 ( (2 x)+a (4 x)-b)/(x^4) is finite, then (a+b) is equal to : JEE Main 2025 Full Limits list Limits JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs