Harmonic series

Even a pile of iden­tical bases can brighten up learning math­e­matics and physics.

To what extent can one move an upper base, so that it won’t fall? We know the answer from our child­hood: to the half of length — so that gravity centre doesn’t go out of the other layers’ base­ment.

And now — to what extent can one move two upper layers? In this case an exper­i­ment should be proved by calcu­la­tion. It’s inter­esting to count, where a gravity centre of maxi­mally moved several bases is located.

And how far can such “ladder” go hori­zon­tally? Math­e­mat­i­cally, it can go infi­nitely, as maxi­mally possible moves of the bases corre­spond to a diver­gent harmonic row. In the movie Stairway to infinity a mechan­ical loader and a tractor move bricks in an optimal way and explain the proof of a harmonic row’s diver­gence. And one can make an exper­i­ment with one’s friends, how far can one build such a ladder.