pi formula

Taylor-made Pi

After learning about the Taylor series for 1/(1+x) in calculus, you can find an interesting expression for Pi very easily. Start with 1/(1+w) = 1 – w + w2 – w3 + … Now substitute x2 for w: 1/(1+x2) = 1 – x2 + x4 – x6 + … Then integrate both sides...

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Pi Approximations

Pi is the ratio of the circumference of a circle to its diameter. It is known to be irrational and its decimal expansion therefore does not terminate or repeat. The first 40 places are: 3.14159 26535 89793 23846 26433 83279 50288 41971… Thus, it is sometimes helpful...

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Buffon Needle Problem

A plane is ruled with parallel lines 1 cm apart. A needle of length 1 cm is dropped randomly on the plane. What is the probability that the needle will be lying across one of the lines? Answer: 2/Pi. This gives an interesting way to...

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Wallis’ Formula

Here’s a fun formula for Pi involving an infinite product, known as Wallis’ Formula: (Pi/2) = (2*2)(4*4)(6*6)/(1*3)(3*5)(5*7) It is somewhat surprising that when you pull out every other pair of terms, you get a completely different kind of number! Sqrt[2] = (2*2)(6*6)(10*10)/(1*3)(5*7)(9*11) Presentation Suggestions:You can also start with...

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