Endless series of rectangles

The area of the first $k$ such rectangles is

$\begin{aligned}1\cdot\sum_{n=1}^{k}\left [\frac{1}{n}-\frac{1}{n+1}\right ]&=1+\left (-\frac{1}{2}+\frac{1}{2}\right )+\left (-\frac{1}{3}+\frac{1}{3}\right )+...+\frac{1}{k-1}+\left (-\frac{1}{k}+\frac{1}{k}\right )-\frac{1}{k+1}\\&=1-\frac{1}{k+1}\end{aligned}$

As $k\to\infty$ , $\frac{1}{k+1}\to 0$ , so the total area of the rectangles is $1-0=\color{#20A900}{\boxed{1}}$ .

The area is given by

$\begin{aligned} A & = 1 \times \left(\frac 11 - \frac 12 \right) + 1 \times \left(\frac 12 - \frac 13 \right) + 1 \times \left(\frac 13 - \frac 14 \right) + \cdots \\ & = 1 - \red{\cancel{\frac 12}} + \blue{\cancel{ \frac 12}} - \red{\cancel{\frac 13}} + \blue{\cancel{ \frac 13}} - \red{\cancel{\frac 14}} + \blue{\cancel{ \frac 14}} - \cdots \\ & = \boxed 1 \end{aligned}$