I Conflicting Conventions for Bernoulli Numbers?

AI Thread Summary
The discussion centers on the differing conventions for Bernoulli numbers as outlined in the Wikipedia article. It highlights that the two conventions differ only at m=1, with the implication that for m>1, the expressions are equivalent despite additional terms in one expression. Participants clarify that the cancellation of terms is indeed valid due to the properties of binomial coefficients following Pascal's triangle. This understanding reassures that the perceived complexity is not a misinterpretation. The conversation concludes with confirmation of the mathematical principles involved.
nomadreid
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In the Wiki article on Bernoulli numbers, it gives two expressions that, if I understand correctly, are supposed to be equal except at one point. But I am not sure I understand it correctly
In the Wikipedia article https://en.wikipedia.org/wiki/Bernoulli_number on Bernoulli’s numbers, it explains that there are two conventions which differ only at m=1. Then it says…

Bernoulli1.PNG


Under “explicit definitions”, it gives, for m>1

Bernoulli2.PNG


So, it seems pretty straightforward that they are saying that (except for m=1) these two expressions are equal, but that all the extra terms in the second expression (+) not included in the first one (-) would cancel out seems so incredible that I think I might be misinterpreting something. Am I?

Thanks
 
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The expansion of ##(v+1)^m## follows Pascal's triangle, hence binomial coefficients, so yes, the terms can cancel out.
 
Thanks, DrClaude
 
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