Here is an old but interesting task from a math book, but I'm completely clueless:
A teacher stands at the chalkboard and draws a line, starting from the left side of the board, towards the right. The line is ten centimeters long. Then he draws, without interruption, a second line with half of the length of the first line towards the right. Then a third, with a third of the length of the first. Then a fourth with a quarter of the length, and so on and so forth. The teacher works very carefully. With pointed chalk, he adds a new line per second. The chalkboard is one meter and ninety centimeters wide.
How long the teacher needs, to arrive at the right side of the chalkboard?
As the teacher finished, he constructed a square on each line. The side length of the squares corresponds to the length of the respective lines.
How big is the surface area of all the squares together, approximately?
Can't even tell if it is about a convergent respectively divergent series.
Can you help? Thank you!
This is indeed a question about a convergent / divergent series.
After n seconds, the teacher will have drawn $\sum\limits_{i=1}\limits^{n}\frac{0.1}{n}$ meters of line. $\sum\limits_{i=1}\limits^{\infty}\frac{1}{n}$ is called the harmonic series and it is divergent.
Numerically, I get that if n is around 100000000, then $\sum\limits_{i=1}\limits^{n}\frac{0.1}{n} \approx 1.9 $ meters.
The sum of the surface areas however, $\sum\limits_{i=1}\limits^{n}\left(\frac{0.1}{n}\right)^2$ is a convergent series. With alot of effort, one can show that the series $\sum\limits_{i=1}\limits^{\infty}\left(\frac{1}{n}\right)^2 = \frac{\pi^2}{6}$.
The lesson to be learend here is probably that if you wait long enough, the line is going to become arbitrarily long, while surprisingly the sum the area of the squares is never going to exceed $\frac{0.1^2 \cdot \pi^2}{6}$.