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Topic: Bernoulli numbers and sqrt(1)+sqrt(2)+sqrt(3) + ... sqrt(1000)
Replies: 9   Last Post: Feb 18, 2013 2:34 PM

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David Bernier

Posts: 3,892
Registered: 12/13/04
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:

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

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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