The Sublime & Beautiful vs. Reality

This blog is a record of one man's struggle to search for scientific, philosophical, and religious truth in the face of the limitations imposed on him by economics, psychology, and social conditioning; it is the philosophical outworking of everyday life in contrast to ideals and how it could have been.


-
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God
and which He revealed to us in the language of mathematics.
--Johannes Kepler

___


|
--(:|:)--
|
|
___________________________________________

Monday, January 16, 2006

Personal Update: Running ahead of my studies

Bernoulli numbers were defined by Jacob Bernoulli in connection with evaluating sums of the form ∑i^k.
The sequence B0, B1, B2, ... can be generated using the formula
x/(e^x - 1) =∑(Bn x^n)/n!
though various different notations are used for them.
The first few are: B0 = 1 , B1 = -1/2 , B2 = 1/6 , B4 = -1/30 , B6 = 1/42 , ...
They occur in many diverse areas of mathematics including the series expansions of tan(x), Fermat's Last theorem†, etc.


†Fermat's last theorem states that if n > 2 then the equation x^n + y^n = z^n has no positive integer solutions.

(Please note that in the second summation it is B sub-n times x^n, etc.)
It looks like I'll have to start a separate web page just to be able to publish mathematics symbols--this text editor for the blog is rather limited.

No comments: