Fw: e 為無理數

[Dirichlet (微風輕吹)]

※ [本文轉錄自 Dirichlet 信箱]

作者: [e-mail] ("Weak and only weak")
標題: e 為無理數
時間: Sat Jul 19 21:43:58 2014

作者: pretend (Big Lion) 站內: pretend
標題: e 為無理數
時間: Fri Nov  5 07:38:38 2004


                                      oo
Assume e is a rational number ==> e = Σ 1/k! = q/p , gcd(p,q)=1 , p,q belong
                                      k=0
                       p
to N . Hence e(p!) = (Σ p!/k!) + [p!/(p+1)! +p!/(p+2)! +...] = q[(p-1)!]
                      k=0

But [p!/(p+1)! +p!/(p+2)! +...]  <  p!/(p+1)! [1 +1/(p+1) + 1/(p+1)^2 + ...]

                1
= p!/(p+1)! --------- = 1/p < 1
             p/(p+1)

                     p
It is obvious that (Σ p!/k!) + [p!/(p+1)! +p!/(p+2)! +...] ≠ q[(p-1)!]
                    k=0

So e is a irrational number ......



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※ 轉錄者: Dirichlet (111.255.113.174), 07/19/2014 21:44:36