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I have to solve:

$$\lim_{n\to\infty} \frac{\sum_{k=1}^{n}n^p}{n^{p+1}}$$ where $p \in \mathbb{R}$

So far I did the basics but don't know what to do from here:

$$ \lim_{n\to\infty} \frac{(n+1)^p}{(n+1)^{p+1}-n^{p+1}}$$

i_suck_at_math
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  • You could apply the theorem $p$ more times. Take into account that applying the finite difference to a polynomial reduces its degree, and multiplies its leading term by the degree, similar to a derivative. After $p$ applications of the theorem you are left with computing the limit of a quotient of two constants. – conditionalMethod Nov 07 '19 at 15:42
  • See also https://math.stackexchange.com/questions/150391/evaluate-lim-limits-n-to-infty-frac-sum-k-1n-kmnm1 – Arnaud D. Nov 07 '19 at 15:48

1 Answers1

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Recall by binomial theorem that

$$(n+1)^{p+1}-n^{p+1}\sim (p+1)n^p$$

user
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