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Wed 14 Apr, 2004 06:16 pm
I remember seeing this somewhere, don't know where, don't know when, but it went something along the lines of
Pi/4 = 1-1/3+1/5-1/7+1/9...
I realise that Pi/4 = 11/14, but does anyone know what this is about? And how the hell does someone get to discover this...??
Thanks for that - I'll have a search around and find out a bit more now I know what I'm aiming for.
Is that leibniz's only formula?!
If you multiply each term by x raised to the power of the denominator, it's the cosine of x. Dunno how to get from there to pi/4, though. Hmmmm....
patiodog wrote:If you multiply each term by x raised to the power of the denominator, it's the cosine of x. Dunno how to get from there to pi/4, though. Hmmmm....
Actually, it is not cos(x) but arctan(x) (inverse tangent):
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...
sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...
cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...
To see that arctan(1) equals pi/4, just apply the definition of arctan to an isosceles right (45-45-90) triangle.
http://mathworld.wolfram.com/RightTriangle.html
http://mathworld.wolfram.com/Trigonometry.html
http://mathworld.wolfram.com/InverseTangent.html
Oh yea, speaking of series and such...can someone tell me the names of the most commonly used series in Calculus? Thanks.
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