Calculus proof help

• Jan 25th 2009, 12:20 AM
wyhwang7
Calculus proof help
Hi! I have a midterm on Monday which will be based on homework, and there is ONE problem that I am having difficulty answering.

Which is..

Use the following equation: * <= * means greater than or equal to

e^(1/(n+1)) <= 1 + 1/n <= e^(1/n)

to prove that for n>0,

(1 + 1/n)^n <= e <= (1 + 1/n)^(n+1)

This is part of the section in my calculus book called "Compound Interest and Present Value"

Thanks!
• Jan 25th 2009, 01:04 AM
running-gag
Hi

Starting from
$\displaystyle e^{\frac{1}{n+1}} \leq 1+\frac{1}{n} \leq e^{\frac{1}{n}}$

Take the natural logrithm (which is an increasing function therefore does not change the sense of inequalities)
$\displaystyle \frac{1}{n+1} \leq ln\left(1+\frac{1}{n}\right) \leq \frac{1}{n}$ (1)

Now multiply (1) by n
$\displaystyle n\: ln\left(1+\frac{1}{n}\right) \leq 1$

$\displaystyle ln\left({1+\frac{1}{n}}\right)^n \leq 1$

And apply exponentiation (which is an increasing function therefore does not change the sense of inequalities)

$\displaystyle \left({1+\frac{1}{n}}\right)^n \leq e$

Multiply (1) by (n+1) instead of n to get the second part of the inequality you are looking for
• Jan 25th 2009, 11:51 AM
wyhwang7
Thank you so much!