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Bernoulli numbers and sqrt(1)+sqrt(2)+sqrt(3) + ... sqrt(1000)
Posted:
Feb 16, 2013 1:42 AM
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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?
My first thought would be that the Euler-MacLaurin
summation method might be applicable.
Above, if k^a is the k'th term, a = 1/2 .
There's also a= -1 that leads to partial sums of
the harmonic series.
David Bernier
--
dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.
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