I am currently studying the integral test for infinite series, but I am confused by the pictures used in the integral tests to show that $\sum \frac{1}{n}$ diverges, but $\sum \frac{1}{n^2}$ converges.
The explanation for the divergence of $\frac{1}{n}$ uses this picture of $\frac{1}{x}$ to show that $\sum \frac{1}{n} \geq \int \frac{1}{x} dx$, so that $\sum \frac{1}{n}$ diverges. Now, for the convergence of $\sum \frac{1}{n^2}$ this picture of $\frac{1}{x^2}$ is used to show $\sum \frac{1}{n^2} \leq 1 + \int \frac{1}{x} dx$. However, in the first picture the rectangles are drawn from the upper left corner, while the upper right corner is used in the second picture. If we were to draw the rectangles from the left corner, the rectangles would be shifted to the left by 1 so that all recantgles are less than the integral at each point (i think)
So, my question is how to decide what corner to use to be able to determine whether the value of the integral will be less than or greater than the sum of the rectangles.
It really depends on what you want to prove by the comparison test.
Rectangles drawn with the upper left corner on the curve result in a sum that is greater than the integral, so if the integral diverges, the sum also diverges. But if the integral converges, no information on the sum is obtained.
Rectangles drawn with the upper right corner on the curve result in a sum that is less than the integral, so if the integral converges, the sum also converges. But if the integral diverges, no information on the sum is obtained.