Previously, we introduced the Bernoulli polynomials, and the Bernoulli numbers. In particular, we found that the Bernoulli numbers obey (and along with *B*_{0}=1 can be defined by) the relation .

Now, let us examine the Maclaurin series for the function (the singularity at *z*=0 is removable, and is also removable in the derivatives of all order).

Let , so that our series is then

Now, the Maclaurin series for *e*^{x} is , so the series for *e*^{z}-1 is

Thus, we have:

.

Now, we use the Cauchy product for infinite series:

, where .

This tells us

,

where

But the product of the two series is unity, so we have *c*_{0}=1, *c*_{n}=0 for *n*≥1. Thus

and

for *n*≥1

This, however, is the relation for the Bernoulli numbers: , and so the Maclaurin series is

,

and .

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Tags: Bernoulli Numbers, Generating Function, Maclaurin Series, Math, Monday Math, Taylor Series

This entry was posted on February 2, 2009 at 4:13 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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February 9, 2009 at 12:36 am |

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