
Bernoulli numbers and sqrt(1)+sqrt(2)+sqrt(3) + ... sqrt(1000)
Posted:
Feb 16, 2013 1:42 AM


The Bernoulli numbers can be used to compute for example 1^10 + 2^10 + ... + 1000^10 .
Jakob Bernoulli wrote around 17001713 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 EulerMacLaurin 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.

