if f(1) + f(2) +... + f(n-1)< the integral from 1 to n of f(x)dx < f(2) + f(3) +...+ f(n).
Chose f(x) = ln x. Show that n^n/e^(n-1)< n! < n^(n+1)/e^n
Assume that and
And
As well as
So we would have that
And since the exponential function is stricly increasing across its domain the above inequality also gives
So then because is strictly increasing we may rewrite our inequality as
Reciporcating and not forgetting to reverse the direction of the inequalities gives
Multiplying through by gives
Finally multiplying through by gives