If _ n → ∞ (n + 1) ^ k - 1 n ^ k + 1 [( nk + 1 ) + ( nk + 2 ) + +...

Given,

$\lim_{n \to \infty}\dfrac{{\left(n + 1\right)}^{k - 1}}{{n}^{k + 1}}\left[( nk + 1 ) + ( nk + 2 ) + \ldots + \left(nk + n\right)\right] = 33$

Taking $LHS$

$\lim_{n \to \infty}\dfrac{{\left(n + 1\right)}^{k - 1}}{{n}^{k + 1}}\left[nk \cdot n + 1 + 2 + \ldots + n\right]$

$= \lim_{n \to \infty}\dfrac{{\left(n + 1\right)}^{{}^{k - 1}}}{{n}^{k + 1}} \cdot \left[{n}^{2}k + \dfrac{n\left(n + 1\right)}{2}\right]$

$= \lim_{n \to \infty}\dfrac{{\left(n + 1\right)}^{k - 1} \cdot {n}^{2}\left(k + \dfrac{\left(1 + \dfrac{1}{n}\right)}{2}\right)}{{n}^{k + 1}}$

$= \lim_{n \to \infty}\left(1 + \dfrac{1}{n}\right)\left(k + \dfrac{\left(1 + \dfrac{1}{n}\right)}{2}\right)$

$= \left(k + \dfrac{1}{2}\right)$

Now taking $RHS$

$= 33\lim_{n \to \infty}\dfrac{1}{{n}^{k + 1}}\left({1}^{k} + {2}^{k} + \ldots + {n}^{k}\right) = 33\lim_{n \to \infty}\mathop{∑}_{k = 1}^{n}\dfrac{1}{n}{\left(\dfrac{k}{n}\right)}^{k}$

$= 33{∫}_{0}^{1}{x}^{k}dx = \dfrac{33}{k + 1}$

Now given, $LHS = RHS$

$⇒ k + \dfrac{1}{2} = 33 \cdot \dfrac{1}{k + 1}$

$⇒ \left(2k + 1\right)\left(k + 1\right) = 66$

$⇒ \left(k - 5\right)\left(2k + 13\right) = 0$

$⇒ k =5$ or $- \dfrac{13}{2}$

So, integral value of $k = 5$