
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!

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
