Is there a proof that π is an irrational number?

Question

Is there a proof that   π is an irrational number?

Asdrubali2r · Accepted Answer

Let       I    n    (  α  )  =      ∫          −      1        1    (  1  −      x    2        )    n    cos  ⁡  α  x       d    xthen 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  )  &amp;lt;      ∫          −      1        1    (  1  −      x    2        )    n         d    x  &amp;lt;  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.

Is there a proof that π

Answered question

Answer & Explanation

New Questions in Algebra I