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$$\lim_{n\rightarrow \infty} \frac{\sum_{i=1}^{n}{i^{p}}}{n^{p}}-\frac{n}{p+1}$$ I know that $$\lim_{n\rightarrow \infty} {x}_{n}= \frac{1}{2}$$ but how can i prove it? I've seen it, by it's very hard

Evaluating $\lim\limits_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$

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Viktor
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    What can you throw at it? If you consider the sum as a Riemann sum, it's not hard. – Daniel Fischer Dec 04 '13 at 22:28
  • @DanielFischer That's the first thing I thought of when I saw this problem. But it wasn't so straight forward when I tried it. Could you explain the method a little bit? – Pratyush Sarkar Dec 05 '13 at 04:17
  • @PratyushSarkar One way is Did's answer to the linked question. You can also estimate the difference between the sum and the integral in other ways, trapezoidal rule, for example. It's not totally obvious, but it's not hard to follow the estimates. – Daniel Fischer Dec 05 '13 at 09:21

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