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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 228

I have developed new method for Numerical Analysis for sums of power for arithmetic progression of non-integer power and still working on it to improve its accuracy. However, before I could present it, has anybody got an idea how to sum this series

. Is there any formulation that could sum it?*Last edited by Stangerzv (2012-07-19 14:11:17)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,365

Hi Stangerzv;

There is a closed form for that sum but it is not a simple one. The answer involves the Hurwitz zeta function.

There are numerous ways in numerical analysis to do sum approximately.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 228

Thanks bobbym for the info. I have developed two ways of finding the sums, first method is using Area Integration which gives quite high error. Another one is using unbounded sums of power of arithmetic progression. After using 4 internal coefficients I managed to get 0.004% Error. As the internal coefficients reach infinity the sum will approach 100% accuracy. Can you show me the closed form formulation? As to my knowledge, if the closed form for this series existed then the sums of power for arithmetic progression would be technically found ages ago. There is a paper on the sums of power formulation by Chen et al dated 2008 and accepted by 2010, you can read their paper by the title Faulhaber's Theorem on Power Sum at arxiv, he actually managed to formulate for odd power.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,365

Hi;

A first, crude bound on the sum is

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 228

Thanks for the formula bobbym! Using mathcad, you would get the value of

. Using method of area integration, , where "a" is the first term and "b" the final term . My method is simple and giving better accuracy. I would write it down later on.*Last edited by Stangerzv (2012-07-21 13:46:04)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,365

Hi Stangerzv;

The replacing of a sum with an integral will yield approximations of the sum. There are much better ways to produce better estimates.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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