[大學]exp(x)相關證明

[大學]exp(x)相關證明

Jimese 於 星期日 五月 23, 2021 9:22 pm


Prove, if n is a positive integer, then there exists N such thate^x > x^nfor all x ≥ N.

Jimese
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Re: [大學]exp(x)相關證明

lskuo 於 星期二 五月 25, 2021 1:44 pm


Jimese 寫到:Prove, if n is a positive integer, then there exists N such that e^x > x^n for all x ≥ N.



This is a very classic property of an exponential function. You should be able to find it out in any textbook of calculus.

Hint:
First show that e^x > (1+x) for x > 0
Then let f_1(x) = e^x - (1+x). Again show that f_1(x)>0 for x>0
Continue this way,
  f_n(x) = e^x - (1+x + x^2/2! + x^3/3! + ... + x^n / n!). Show that f_n(x) > 0 for x>0.

By this way, you can show that
   e^x > 1 + x + x^2/2! + ... + x^(n+1)/(n+1)!

Dividing two sides by x^n,

  e^x / x^n > x/(n+1)! for x > 0

Now you know how to estimate N.

lskuo
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