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: Re: e 為無理數
時間: Sat Jul 19 21:44:02 2014

作者: pretend (Big Lion) 站內: pretend
標題: Re: e 為無理數
時間: Fri Nov  5 07:48:14 2004

※ 引述《pretend (Big Lion)》之銘言：
>                                    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 ......

note : e > 2 is trivial ....
       e= 1 + 1 + 1/2! + ... < 1 + 1 + 1/2!(1 + 1/3 + 1/3^2 + ...)
        = 1 + 1 + (1/2!)(3/2) = 2 + 3/4 < 3
       Since 2 < e < 3 , it follows that p ≠ 1

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※ 轉錄者: Dirichlet (111.255.113.174), 07/19/2014 21:45:56