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.