I have the following problem and I am stuck on it. Any ideas on how to work it?

Let k be a positive integer. Show that 1^k + 2^k + ... + n^k is O( n^(k+1) )

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- Mar 19th 2009, 10:08 AMothninBig O puzzler
I have the following problem and I am stuck on it. Any ideas on how to work it?

Let k be a positive integer. Show that 1^k + 2^k + ... + n^k is O( n^(k+1) ) - Mar 20th 2009, 06:36 AMOpalg
You want to show that $\displaystyle \frac{\sum_{r=1}^nr^k}{n^{k+1}}$ is bounded as $\displaystyle n\to\infty$. One way is to write that expression as $\displaystyle \frac1n\sum_{r=1}^n\Bigl(\frac rn\Bigr)^k$ and think of it as a Riemann sum approximating $\displaystyle \int_0^1\!\!x^k = \frac1{k+1}$.

- Mar 20th 2009, 12:33 PMothnin
Interesting,

I was trying to do a proof by Induction. Do you know if it's possible to it that way?

Thanks for heads up! - Mar 21st 2009, 03:52 AMLaurent
- Mar 21st 2009, 02:22 PMothnin
Thanks guys. I think I can get a proof out of it now.