Improper integral divergence

Question

Use the graph of 1/x and the sum of areas of rectangles to show that $\int _{ 1 }^{ \infty }{ \frac { 1 }{ x } dx }$ = +$\infty$.

Would the sum of rectangles just be:

1 + 1/2 + 1/3 + 1/4 +....+1/n + = +$\infty$.

Answer 1

You will need to draw a picture. Then maybe use:

First rectangle: base $[1,2]$, height $1/2$;

Second rectangle: base $[2,4]$, height $1/4$;

Third rectangle: base $[4,8}$, height $1/8$;

Fourth rectangle: base $[8,16]$, height $1/16$;

And so on.

Note that each rectangle has area $1/2$, and the union of the rctangles lies in the region "below" $y=1/x$ and "above" the $x$-axis.