Difference between revisions of "E"
(An interesting formular relating e and pi by ramunajan) |
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<math>e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots</math> | <math>e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots</math> | ||
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As an infinite sum it is ''Σ(x<sup>n</sup>/n!)'' = 1 | As an infinite sum it is ''Σ(x<sup>n</sup>/n!)'' = 1 | ||
Revision as of 08:14, 28 August 2006
| A page for e, i, π, and φ |
The role that the exponential function (ex) plays is extremely significant to an understanding of the laws of nature and harmonic organisation.
[math]e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots[/math]
As an infinite sum it is Σ(xn/n!) = 1
+ x / 1
+ x.x / 1.2
+ x.x.x / 1.2.3
+ x.x.x.x / 1.2.3.4 + ...
We can look at the construction of this relation more easily by describing it by an algorithm. We can remove the power and factorial operations by basing the current numerator and denominator values on the previous values:
| +e.pl |
But a better conceptual representation is to restructure it such that the division occurs outside the main loop since its also a higher-level operation. So each iteration is now maintaining the value of a numerator and denominator which represent the entire preceeding series as a single fraction which can be reduced afterwards.
| +alt e.pl |
Here's a snipit in C which uses the taylor series of cosine to make Φ from π
| +phi.c |
- See also
