Mentallic
Oct27-09, 01:04 AM
1. The problem statement, all variables and given/known data
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 e^{-\frac{n}{n-1}}<\left(1-\frac{1}{n}\right)^n<e^{-1}
3. The attempt at a solution
The area under the curve is more than the smaller rectangle but less than the larger rectangle.
Thus, \frac{1}{n}<\int^n_{n-1}\frac{dx}{x}<\frac{1}{n-1}
After manipulating somewhat:
\frac{1}{n}<ln\left(\frac{n}{n-1}\right)<\frac{1}{n-1}
e^{\frac{1}{n}}<\frac{n}{n-1}<e^{\frac{1}{n-1}}
but I'm unsure how to get to the answer...
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 e^{-\frac{n}{n-1}}<\left(1-\frac{1}{n}\right)^n<e^{-1}
3. The attempt at a solution
The area under the curve is more than the smaller rectangle but less than the larger rectangle.
Thus, \frac{1}{n}<\int^n_{n-1}\frac{dx}{x}<\frac{1}{n-1}
After manipulating somewhat:
\frac{1}{n}<ln\left(\frac{n}{n-1}\right)<\frac{1}{n-1}
e^{\frac{1}{n}}<\frac{n}{n-1}<e^{\frac{1}{n-1}}
but I'm unsure how to get to the answer...