<s>[INST] Find $ \lim_{k \rightarrow \infty } \frac{ \sum_{x = 0}^{k} {x^{n + 1}}} {(k + 1)k^{n + 1}}$ How to compute the limit 
$$ \lim_{k \rightarrow   \infty  } \frac{ \sum_{x = 0}^{k} {x^{n + 1}}} {(k + 1)k^{n + 1}}$$
Does exist an explicit formula for finding the numerator?
 [/INST] By the Stolz-Cesro Theorem , one has
\begin{eqnarray}
&&\lim_{k \rightarrow   \infty  } \frac{ \sum_{x = 0}^{k} {x^{n + 1}}} {(k + 1)k^{n + 1}}\\
&=&\lim_{k \rightarrow   \infty  } \frac{{(k+1)^{n + 1}}} {(k + 2)(k+1)^{n + 1}-(k + 1)k^{n + 1}}\\
&=&\lim_{k \rightarrow   \infty  } \frac{1} {(k + 2)-(k + 1)\bigg(\frac{k}{k+1}\bigg)^{n + 1}}\\
&=&\lim_{k \rightarrow   \infty  } \frac{1} {(k + 2)-(k + 1)\bigg(1-\frac{1}{k+1}\bigg)^{n + 1}}\\
&=&\lim_{k \rightarrow   \infty  } \frac{1} {(k + 2)-(k + 1)\bigg(1-(n+1)\frac{1}{k+1}+O(\frac1{(k+1)^2})\bigg)}\\
&=&\frac 1{n+2}.
\end{eqnarray}
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