How Can I Prove (1+x/n^n < e^x)?

[wowolala]

the quesion is below, show that
show that ( 1+ $$\frac{x}{n}$$ ) ^n < e^x

at the first, i take log on both sides,.. but i couldn't go further.

can someboday help me?

thx

 

[EnumaElish]

What do you get when you apply log to both sides?

 

[mathman]

the quesion is below, show that
show that ( 1+ $$\frac{x}{n}$$ ) ^n < e^x

at the first, i take log on both sides,.. but i couldn't go further.

can someboday help me?

thx

Expand the lhs by the binomial theorem and take the power series for e^x. The solution becomes obvious for x>0. For x<0, it is a little trickier - work with terms 2 at a time.

 

[wowolala]

thx so much