It is well-known that π is irrational, but proving that π is irrational is a surprisingly tricky thing to do. There's a really good video on YouTube that walks through a proof of this.
#MathSky #Maths #Math
#MathSky #Maths #Math
Comments
In particular, integration by parts plays a central role.
- for every n, f_n(0), f_n'(0), f_n''(0), etc, f_n(pi), f_n'(pi), f_n''(pi), etc, are all integers
- for every n, f_n(x) is positive for x in the interval (0,pi)
- the maximum value of f_n(x) on [0,pi] goes to zero as n goes to infinity
Integral_{x=0..pi} sin(x)*f_n(x) dx
is an positive integer for every n (this comes from repeated integration by parts), and that it goes to zero as n goes to infinity.
This is impossible, so no such sequence of f_n's can exist.
f_n(x) = B^n x^n (pi-x)^n / n!
satisfies all three conditions.
B^n x^n (A/B-x)^n / n!
is this optimal polynomial.