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How to show that $e^{x} \geq \left (1 + \frac{x}{n} \right) ^{n}$ holds for each non-negative real $x$ and each integer $n \geq 1$ ? I tried series and induction but got stuck. Can you please help?

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Where did you get stuck exactly? Also, if this is homework, please consider tagging it as homework. –  Aryabhata Nov 4 '10 at 17:14
    
@Moron: Perhaps even i could have asked that question...:x)! –  anonymous Nov 4 '10 at 17:21
    
Hint : use, $\exp(y) \geq 1 + y$ for all $y$. –  ACARCHAU Nov 4 '10 at 17:27
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2 Answers

HINT $\ $ Consider $\rm\ e^z\ \ge\ 1 + z,\ \ z\ =\ x/n $

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Thanks, this works nicely. –  student Nov 4 '10 at 18:24
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For each fixed nonnegative $x$, $(1+\frac{x}{n})^n$ increases with $n$. To see why this is true, note that $(1+\frac{x}{n})^n$ is the amount of money in an account one year after $1$ unit of money is invested at a yearly interest rate of $100x\%$, compounded $n$ times per year. As $n$ goes to $\infty$, this increasing sequence converges to $e^x$, corresponding to continuously compounded interest.

(I do not claim that this is a rigorous proof. It is merely intended as an explanation.)

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