# Thread: Show that ln5 < 1 + 1/2 + 1/3 + 1/4.

1. ## Show that ln5 < 1 + 1/2 + 1/3 + 1/4.

Show that $\displaystyle ln5 < 1 + 1/2 + 1/3 + 1/4.$ using integrals

I guess....First I have to show that the integral from 1 to 5 of $\displaystyle \frac {1}x$ is equal to $\displaystyle ln(5)$.

Then estimate the integral by a finite sum of left endpoints.

Since 1/x is decreasing from 1 to 5, the finite sum I get will be strictly greater than the integral.

But I don't know how to show the steps....

How do I show that the integral from 1 to 5 of $\displaystyle \frac {1}x$ is equal to $\displaystyle ln(5)$?

How do I estimate the integral by a finite sum of left endpoints?

2. Originally Posted by cammywhite
How do I show that the integral from 1 to 5 of $\displaystyle \frac {1}x$ is equal to $\displaystyle ln(5)$?
Evaluate $\displaystyle \int_1^5\frac{dx}x$ using the fundamental theorem of calculus.

How do I estimate the integral by a finite sum of left endpoints?
Add up the areas of the four rectangles. In each subinterval, $\displaystyle \Delta x=1$ and $\displaystyle f(c)=\frac1c,$ where $\displaystyle x=c$ is the left endpoint of the subinterval.

3. ## details of proof

see attachment