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Re: Bernoulli numbers and sqrt(1)+sqrt(2)+sqrt(3) + ... sqrt(1000)
Posted:
Feb 18, 2013 2:26 AM
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Am 16.02.2013 07:42 schrieb David Bernier:
> The Bernoulli numbers can be used to compute for example
> 1^10 + 2^10 + ... + 1000^10 .
>
> Jakob Bernoulli wrote around 1700-1713 that he had computed
> the sum of the 10th powers of the integers 1 through 1000,
> with the result:
> 91409924241424243424241924242500
>
> in less than "one half of a quarter hour" ...
>
> Suppose we change the exponent from 10 to 1/2, so the sum
> is then:
> sqrt(1) + sqrt(2) + ... sqrt(1000).
>
> Or, more generally,
> sqrt(1) + sqrt(2) + ... sqrt(N) , N some largish positive
> integer.
>
> Can Bernoulli numbers or some generalization be used
> to compute that efficiently and accurately?
>
I've done an exploration of the integrals of the Bernoulli-
polynomials which I called for convenience Zeta-polynomials
and which I studied as a matrix of coefficients, which I
call "ZETA"-matrix [2]. Each row r gives the coefficients for
the sums of like powers with exponent r, so we get the
polynomials for r=0,1,2,3,... in one aggregate of numbers.
It is then natural to generalize the creation-rule for that
matrix to fractional row-indexes. However, this method
gives then no more polynomials but series (which is not
what you want, sorry...). That series have the form
infty
S_r(a,b) = sum zeta(-r+c) * binomial(r,c) *((a+1)^r - b^r)
k=0
where the definition of the binomials is also generalized
to fractional r (the cases, when -r+c=1 must be handled by
replacing zeta(1)/gamma(0) by +1 or -1, don't recall the required
sign at the moment) It gives then the sum for the r'th powers
from the bases a to b in steps by 1 and for the natural
numbers r give the Bernoulli-polynomials in the original form
of Faulhaber.
If you are happy with approximations like in your examples,
this all will not of much help/inspiration though, I'm afraid..
Gottfried Helms
[1] http://go.helms-net.de/math/binomial_new/
[2] http://go.helms-net.de/math/binomial_new/04_3_SummingOfLikePowers.pdf
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