How to Prove the Inequality e^x > (1 +f(x)/n)^n for x in (0, infinity)?

[regularngon]

Homework Statement

If 0 <= f(x) < infinity, then I need to show that e^x > (1 +f(x)/n)^n for x in (0, infinity)

Homework Equations

The Attempt at a Solution

I'm pretty sure the answer lies in the comparison of the series representation for e^x and writing (1 +f(x)/n)^n out with the binomial theorem. I did so, however I still don't see it.

 

[d_leet]

Is this supposed to be true for all n, or the limit as approaches infinity or what? Suppose f(0)=1 then the statement is false for n=1, and x=0.

 

[regularngon]

for all n, and x > 0 though.

 

[d_leet]

for all n, and x > 0 though.

Again take n=1, and any function f(x) such that f(1)=2, and it is false.

 

[regularngon]

Yea it must be a typo on my teachers part. I'm going to guess he meant e^f(x).

 

[d_leet]

Well in that case writing out the series expansions for both sides of the equation would help.