Thread: 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


Take the natural logrithm (which is an increasing function therefore does not change the sense of inequalities)
(1)

Now multiply (1) by n




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



Multiply (1) by (n+1) instead of n to get the second part of the inequality you are looking for

Thank you so much!