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## Will the real B1 please stand up

The other day I was doing some not so light recreational reading. On page 83 of GAMMA Julian Havil
discusses the Bernoulli numbers. Evidently there exists a set of numbers B(0), B(1), ... such that

sum(k^pwr,k=1..n) = (1/(pwr+1)) * sum((n^i) * B(pwr+1-i)* binomial(pwr+1,pwr+1 - i),i=1..pwr+1)

Havil listed the first 12 Bernoulli numbers B0, B1, ..... = 1 1/2 1/6 0 -1/30 0 1/42 0 -1/30 0 5/66 0, ...

He also stated that the Bernoulli numbers were part of the coefficients of the expansion of x / [ (exp(x) -1)]

I played around with this and got this

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which seemed to agree with Havil for N> 1. Curiously, Havil's B1 = 1/2, while from the expansion B1 = -1/2
I assumed there was a misprint in Havil's book. Then I discovered MAPLE's bernoulli function, and got

>

This strengthened my suspicion that Havil's B1 = 1/2 was an error. But it turned out, to my surprise, that if you try use
Bernoulli numbers to evaluate the sum of powers, Havil's works and MAPLE's doesn't appear to. See below.

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Will the real B1 please stand up. Are you 1/2 or -1/2 ?

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