Limits, Continuity & Differentiability
Riemann Sum — GIF Series to Infinite Sum
nta_pyq_2026_jan
Grade 12
Question:
Let $[\cdot]$ denote the greatest integer function and $f(x)=\displaystyle\lim_{n\to\infty}\frac{1}{n^3}\sum_{k=1}^n\left[\frac{k^2}{3^x}\right]$. Then $12\displaystyle\sum_{j=1}^\infty f(j)$ is equal to _____.
Step-by-Step Solution
Key Concept: For large $n$: $\sum_{k=1}^n\left[\tfrac{k^2}{3^x}\right]\approx\sum_{k=1}^n\tfrac{k^2}{3^x}=\tfrac{1}{3^x}\cdot\tfrac{n(n+1)(2n+1)}{6}\approx\tfrac{n^3}{3\cdot3^x}$. So $f(x)=\tfrac{1}{3^{x+1}}$.
$f(x)=\tfrac{1}{3^{x+1}}$. $12\sum_{j=1}^\infty f(j)=2$.
Correct Answer: 2