Is there a proof that π is an irrational number?

Lorena Beard

Lorena Beard

Answered

2022-07-07

Is there a proof that π is an irrational number?

Answer & Explanation

Asdrubali2r

Asdrubali2r

Expert

2022-07-08Added 14 answers

Let
I n ( α ) = 1 1 ( 1 x 2 ) n cos α x  d x
then integrate by parts to show that for n 2
α 2 I n = 2 n ( 2 n 1 ) I n 1 4 n ( n 1 ) I n 2 .
Use induction to show that for positive integer n we have
α 2 n + 1 I n ( α ) = n ! ( P ( α ) sin α + Q ( α ) cos α ) ,
where P ( α ) and Q ( α ) are polynomials of degree less than 2 n + 1 in α with integral coefficients.
Show that if π / 2 = b / a ,, where a and b are integers, then
b 2 n + 1 I n ( π / 2 ) n ! ( 1 )
would be an integer.
Note that
I n ( π / 2 ) < 1 1 ( 1 x 2 ) n  d x < 2  and  b 2 n + 1 n ! 0  as  n
which results in contradiction since ( 1 ) is supposed to be an integer but we can show that it is as small as one desires.

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