Pi Day, 3.14 approximation

lookagain

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Using  4(113+15 ... ) = π, \displaystyle \ 4\bigg(1 - \dfrac{1}{3} + \dfrac{1}{5} - \ ... \ \bigg) \ = \ \pi, \ you can estimate it to 3.14 without too much effort on a scientific calculator by working out

4[113+ ... +133135+(12)(137)],  \displaystyle 4\bigg[1 - \dfrac{1}{3} + \ ... \ + \dfrac{1}{33} - \dfrac{1}{35} + \bigg(\dfrac{1}{2}\bigg)\bigg(\dfrac{1}{37}\bigg) \bigg], \ \ especially if it has the inverse key,  X1.\displaystyle \ X^{-1}.


If you increase the denominators, and allow the last denominator to be 41, or 45, or 49, for example, then you will get
better results, but I think it will require even more terms on average than before to achieve correct additional digits of pi.
 
I find it amazing that Archimedes already had 3.14, to be exact 3+1071<π<3+10703.14, \text{ to be exact } 3+\dfrac{10}{71} < \pi < 3+\dfrac{10}{70} in 250 B.C. without a calculator!
 
I find it amazing that Archimedes already had 3.14, to be exact 3+1071<π<3+10703.14, \text{ to be exact } 3+\dfrac{10}{71} < \pi < 3+\dfrac{10}{70} in 250 B.C. without a calculator!
He did have an abacus, I believe.

-Dan
 
I am a calculus student, so I attack pi in this way:

0116x16x42x3+4x4 dx=π\displaystyle \int_{0}^{1} \frac{16x - 16}{x^4 -2x^3 + 4x - 4} \ dx = \pi
 
I am a calculus student, so I attack pi in this way:

0116x16x42x3+4x4 dx=π\displaystyle \int_{0}^{1} \frac{16x - 16}{x^4 -2x^3 + 4x - 4} \ dx = \pi
Nice area.
 
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