$$ \begin{aligned} &\text { If } \lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+(n k+n)] \\ &=33 \cdot \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right] \end{aligned}$$, then the integral value of $\mathrm{k}$ is equal to _____________
Answer (integer)
5
Solution
$\lim\limits_{n \rightarrow \infty}\left(\frac{n+1}{n}\right)^{k-1} \frac{1}{n} \sum_{r=1}^{n}\left(k+\frac{r}{n}\right)=33 \lim\limits_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\frac{r}{n}\right)^{k}$
<br/><br/>
$$
\begin{aligned}
&\Rightarrow \int_{0}^{1}(k+x) d x=33 \int_{0}^{1} x^{k} d x \\\\
&\Rightarrow \quad \frac{2 k+1}{2}=\frac{33}{k+1} \\\\
&\Rightarrow \quad k=5
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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