Australian HSC maths extension 2 test question

[Mentallic]

Homework Statement

This question is from the Australian HSC maths extension 2 test. Q8b)

Let n be a positive integer greater than 1.

The area of the region under the curve y=1/x from x=n-1 to x=n is between the areas of two rectangles.

Show that [tex]e^{-\frac{n}{n-1}}<\left(1-\frac{1}{n}\right)^n<e^{-1}[/tex]

The Attempt at a Solution

The area under the curve is more than the smaller rectangle but less than the larger rectangle.

Thus, [tex]\frac{1}{n}<\int^n_{n-1}\frac{dx}{x}<\frac{1}{n-1}[/tex]

After manipulating somewhat:

[tex]\frac{1}{n}<ln\left(\frac{n}{n-1}\right)<\frac{1}{n-1}[/tex]

[tex]e^{\frac{1}{n}}<\frac{n}{n-1}<e^{\frac{1}{n-1}}[/tex]

but I'm unsure how to get to the answer...

 

[lanedance]

try inverting everything as a next step, then consider raising everything to a power

 

[Mentallic]

The reciprocal... how could I miss that... thanks lanedance.