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A Derivation of Wallis's Formula for Pi

For π day plus one, here is a derivation of Wallis's formula for π that is based on probability distributions. Start with the binomial distribution for a fair coin toss. The probability of getting k heads on n tosses is

(nk)12n

Assume an even number of tosses, n=2m. Then the distribution has a maximum at k=m of

(2mm)122m

For large n, the binomial distribution can be closely approximated by a Normal probability density function give by

p(x)=1σ2πexp(xμ)22σ2

where μ=n/2=m and σ2=n/4=m/2. The maximum at x=μ is

p(μ)=1mπ

Comparing this with the maximum for the binomial distribution and taking the limit as m, you get

1π=limm(2mm)m22m

Now we just need the following double factorial identities

(2m)!=(2m1)!!2mm!

2mm!=(2m)!!

Then we can write

(2mm)m22m=(2m1)!!(2m)!!m

and π can be expressed as

π=limm1m((2m)!!(2m1)!!)2

or in product form as

π=limm1mmk=1(2k2k1)2

This is almost Wallis's formula. To get there, note that

mk=1(2k1)2=12m+1mk=1(2k1)(2k+1)

Substitute this into the above formula, take the limit and you get Wallis's formula

π=2k=1(2k)2(2k1)(2k+1)

Is that cool or what?


© 2010-2016 Stefan Hollos and Richard Hollos

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