<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] The required limit is $1/(n+2)$. There probably does not exist a closed formula for the numerator. However, the numerator can be estimated by use of an integral, as follows.
Note that
$$\frac{k^{n+2}}{n+2}=\int_0^{k}x^{n+1}\,\mathrm dx < \sum_{x=0}^kx^{n+1}<\int_1^{k+1}x^{n+1}\,\mathrm dx=\frac{(k+1)^{n+2}}{n+2}-\frac{1}{n+2}$$
Therefore,
$$\frac{k}{(n+2)(k+1)}<\frac{\sum_{x=0}^kx^{n+1}}{(k+1)k^{n+1}}<(1+\frac{1}{k})^{n+1}\frac{1}{n+2}-\frac{1}{(k+1)k^{n+1}(n+2)}$$
and both the r.h.s and the l.h.s tend to $\frac{1}{n+2}$ as $k\to\infty$.
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