JEE Main 2022 (Online) 25th July Morning ShiftQuestion 1
If $\lim_{n\to\infty}\frac{(n+1)^{k-1}}{n^{k+1}}\left[(nk+1)+(nk+2)+\ldots+(nk+n)\right]=33\cdot\lim_{n\to\infty}\frac{1}{n^{k+1}}\left[1^k+2^k+3^k+\ldots+n^k\right]$, then the integral value of $k$ is equal to ________.
Limits & Continuity
Answer$5$
Solution
Concept. Sum the AP on the left; the right side is a Riemann sum of $x^k$. Formulas used. (1) $\sum_{i=1}^n(nk+i)=n^2k+\frac{n(n+1)}2$(2) $\lim\frac1{n^{k+1}}\sum_{i=1}^ni^k=\int_0^1x^k\,dx=\frac1{k+1}$Step 1 - Left. By (1): $$\frac{(n+1)^{k-1}}{n^{k-1}}\cdot\frac{n^2k+\frac{n(n+1)}2}{n^2}\to1\cdot\Big(k+\frac12\Big)$$ Step 2 - Right. By (2): $\frac{33}{k+1}$. Step 3 - Equation. $\big(k+\frac12\big)(k+1)=33\Rightarrow(2k+1)(k+1)=66\Rightarrow2k^2+3k-65=0$. Step 4. $(k-5)(2k+13)=0$: $k=5$ or $-\frac{13}2$. Integer value: $5$. Answer: 5