Proof that 22/7 exceeds π {Pi}

[jay234]

Overview/Background

22⁄7 is a widely used Diophantine approximation of π. It is a convergent in the simple continued fraction expansion of π. It is greater than π, as can be readily seen in the decimal expansions of these values:

The approximation has been known since antiquity. Archimedes wrote the first known proof that 22⁄7 is an overestimate in the 3rd century BC, although he may not have been the first to use that approximation. His proof proceeds by showing that 22⁄7 is greater than the ratio of the perimeter of a circumscribed regular polygon with 96 sides to the diameter of the circle. A more accurate approximation of π is 355/113.
[edit]Basic idea

The basic idea behind the proof can be expressed very succinctly:

Therefore 22⁄7 > π.
The evaluation of this integral was the first problem in the 1968 Putnam Competition.[4] It is easier than most Putnam Competition problems, but the competition often features seemingly obscure problems that turn out to refer to something very familiar.
[edit]Details

That the integral is positive follows from the fact that the integrand is a quotient whose numerator and denominator are both non-negative, being sums or products of powers of non-negative real numbers. Since the integrand is positive, the integral from 0 to 1 is positive because the lower limit of integration is less than the upper limit of integration (0 < 1).
It remains to show that the integral in fact evaluates to the desired quantity:
   
   (expanded terms in numerator)
   (performed polynomial long division)
   (definite integration)
   (use arctan(1) = π/4 and arctan(0) = 0)
   (addition)

Quick upper and lower bounds

In Dalzell (1944), it is pointed out that if 1 is substituted for x in the denominator, one gets a lower bound on the integral, and if 0 is substituted for x in the denominator, one gets an upper bound:[5]

Thus we have

hence 3.1412 < π < 3.1421 in decimal expansion. The bounds deviate by less than 0.015% from π. Also see Dalzell (1971).[6]
[edit]Extensions

The above ideas can be generalized to get better approximations of π. More precisely, for every integer n ≥ 1,

where the middle integral evaluates to involving π. The last sum also appears in Leibniz' formula for π. The correction term and error bound is given by

where the approximation (the tilde means that the quotient of both sides tends to one for large n) follows from Stirling's formula and shows the fast convergence of the integrals to π.
Calculation of these integrals[show]
The results for n = 1 are given above. For n = 2 we get

and

hence 3.14159231 < π < 3.14159289, where the bold digits of the lower and upper bound are those of π. Similarly for n = 3,

with correction term and error bound

hence 3.14159265340 < π < 3.14159265387. The next step for n = 4 is

with

which gives 3.14159265358955 < π < 3.14159265358996.

For complete understanding Assume π = Pi.

[Nigeria Scholar]

@ jay234, why don't u change ur username 2 Mr. Lecture or use d prefix Prof. b4 ur name. Wht do u tink bro?

[ejibond]

Una see y i hate maths?

[jay234]

Una see y i hate maths?

@Ejibond: Hate it or love it.. Mathematics has come to stay! Mathematics is beautiful in its own sense!!... You've got to like it James Bond! Sorry Ejibond..Lol

@ jay234, why don't u change ur username 2 Mr. Lecture or use d prefix Prof. b4 ur name. Wht do u tink bro?

@Chinyxboy: not yet.. Still a beginner in this great ordeal Chinaboy! SOrry Chinyxboy...

[ejibond]

@Ejibond: Hate it or love it.. Mathematics has come to stay! Mathematics is beautiful in its own sense!!... You've got to like it James Bond! Sorry Ejibond..Lol

why i no see dis ontime sef...@jay sean! Oop! I mean jay123456...since i dey sku,dem dey always as us to find value of x,y,a for either simultaneous equation or others abi? Ok! Now tell me,if we don finish sku n get work,how we wan take find those alphabet 4 d work wey we go dey do??

[Nigeria Scholar]

@ ejibond, pls help me ask am ooooo. I tire 4 dis abstract subjects self. Dey jst mak me won 2 fight myself.

[gentleemmy]

Sory, am outta here

[gentleemmy]

Funny enough is very difficult to run away from figures in d den. I was thnkn I escapd bt to my chagrin, I come jam statistics for Geography. Anyways I going to do d best I know on friday wen I face him bt make una warn am coz na him head I go smash

[Nigeria Scholar]

@ gentleemmy, i go like 2 c d scar oooo.

[ejibond]

hahaha!!...na maths make me run frm engineering course cos each time i work maths i go must drink panadol...