Idealistic
Junior Member
- Joined
- Sep 7, 2007
- Messages
- 97
Today in lecture we discussed that with infinite books, you could extend a stack of books on the edge of a table forever by placing the first book on a table, such that it is half off, and continuosly adding more books (positioned as far from the table as posible without tipping the stack) to generate a distance from the the last book added to the stack the the edge of the table.
It was found that the least upper bound for the distance of the stack was found to = Sup(1/(2[sup:2kjbwsga]n[/sup:2kjbwsga])) or half of the harmonic series. Where n is an integer from 1 to infinty (natural numbers) reperesenting the amount of books in the stack.
He stated the forces on either side of the inertia of the first book on the table had to be balanced.
With the length of one book = 1 he had:
the distance (of the first book) off of the table = x
the length of the first book on the table = (1 - x)
the number of books in the stack = n
here's the equation he had:
I'm pretty sure each side of the equation is suposed to represent either side of the inertia (of the first book).
(1 - x)(1 - x)/2 = x(n - 1) + x(x/2) *it simplifies to 1/(2[sup:2kjbwsga]n[/sup:2kjbwsga])
Now im not looking to simplify this my issue is with the explanation of why this is the equation one would use. i.e. why do you multiply (1 - x) by (1 - x)/2? Why do y multiply x by (n - 1) and sum it to x(x/2) ...etc.
Just an explanation if possible would be highly appreciated.
It was found that the least upper bound for the distance of the stack was found to = Sup(1/(2[sup:2kjbwsga]n[/sup:2kjbwsga])) or half of the harmonic series. Where n is an integer from 1 to infinty (natural numbers) reperesenting the amount of books in the stack.
He stated the forces on either side of the inertia of the first book on the table had to be balanced.
With the length of one book = 1 he had:
the distance (of the first book) off of the table = x
the length of the first book on the table = (1 - x)
the number of books in the stack = n
here's the equation he had:
I'm pretty sure each side of the equation is suposed to represent either side of the inertia (of the first book).
(1 - x)(1 - x)/2 = x(n - 1) + x(x/2) *it simplifies to 1/(2[sup:2kjbwsga]n[/sup:2kjbwsga])
Now im not looking to simplify this my issue is with the explanation of why this is the equation one would use. i.e. why do you multiply (1 - x) by (1 - x)/2? Why do y multiply x by (n - 1) and sum it to x(x/2) ...etc.
Just an explanation if possible would be highly appreciated.