Talk:Bernoulli number

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[edit] Inconsistency

Inconsistency here: Bernoulli numbers as defined on Bernouilli number page are alternately negative and positive.

But Taylor series for Tan(x) and Cot(x) use Benouilli numbers that are all positive. Formula for Taylor series should use absolute value |Bn| and not Bn.

Or Bernouilli numbers should be defined as all positive.

There is no inconsistency at all. The article says:
The Bernoulli numbers also appear in the Taylor series expansion of the tangent...
and that is exactly correct; they appear in the Taylor series expansion. The Taylor series for tan(x) is:
\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1}
in which B2n denotes the 2nth Bernoulli number; so the Bernoulli numbers do indeed appear in the Taylor series expansion. The multiplication by (-4)n ensures that all the coefficients are positive. I hope this clears things up for you. -- Dominus 12:40, 29 July 2005 (UTC)

Yes it does, thanks.

Another question:

\sum_{k=0}^{m-1} k^n = 0^n + 1^n + 2^n + \cdots + {(m-1)}^n

for various fixed values of n. The closed forms are always polynomials in m of degree n + 1

But term of highest degree appears to be (m − 1)n which has a degree of 'n'.

there is a sum of m terms involved here. If you take in the sum of powers only those summands say with k   >   m/2− 1 then the cut-off sum will be already bigger than
(m/2) (m/2)^n ={1\over {2^{n+1}}} m^{n+1} \ .
Therefore the order of magnitude (for fixed n) is asymptotically O(mn + 1) and not O(mn) (the exact order is {1\over{n+1}}\,m^{n+1}+O(m^{n}))..

[edit] Big O notation

Thanks for the timely revert, Dmharvey. I mistakenly thought you were using the omega notation in its historical sense, equivalent to big O. Elroch 22:21, 10 February 2006 (UTC)

[edit] Relationship of Bernouilli numbers to Riemann zeta function

I decided to change the language used to describe the relationship of the Bernouilli numbers to the Riemann zeta function, which grated with me as it stood. As I understand it, two sequences are the same "up to a factor" if one is a constant multiple of the other, and describing one sequence as "essentially" another sequence was wooly language at best. Elroch 22:29, 10 February 2006 (UTC)

[edit] Kowa Seki

A recent edit removed the assertion that the Bernoulli numbers were first studied by Bernoulli, and instead attributed them to the great Japanese mathematician Seki "in 1683", and asserting that Bernoulli did not study them until "the 18th Century". The implication here is that Bernoulli was greatly anticipated by Seki. But Bernoulli (1654-1705) and Seki (1642-1708) were contemporaries, and without two dates, I am reluctant to believe any claims of priority.

If anyone has any real information, I would be glad to hear it. Meantime, I am going to change the article again to note Seki's discovery. -- Dominus 01:15, 28 March 2006 (UTC)

[edit] Recursive Definition

I could be mistaken, but I don't see the recursive definition as being recursive. Maybe the 0 on the right hand side should be Bm+1? Psellus 23:19, 7 July 2006

It's recursive but not phrased in an explicitly recursive manner. For example, try substituting m = 3 and then solve for B3. If you like you can rearrange the equation to show the recursion more explicitly, but it's quite elegant the way it's written currently. Dmharvey 23:48, 7 July 2006 (UTC)

I was afraid it would turn out to be something like this. OK, I will look at it harder. Thanks very much. Psellus 23:54, 7 July 2006 (UTC)

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