There is no paradox here. Only a complete misunderstanding of math by some that makes them think there is a paradox.
There is no reason to get hung up on adding shorter and shorter intervals, each 1/2 of the previous interval. It is simple to show that the sum of 1/2 + 1/4 + 1/8 + 1/16 + ... = 1.
Let x = 1/2 + 1/4 + 1/8 + 1/16 + ...
Then 2x = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
Subtract the first from the second and we will get
2x - x = 1 + 1/2 + 1/4 + 1/8 + 1/16 +... - 1/2 - 1/4 - 1/8 - 1/16 - ...
x = 1 + 1/2 - 1/2 + 1/4 - 1/4 + 1/8 - 1/8 +1/16 - 1/16 + ...
= 1.
Thus 1/2 + 1/4 + 1/8 + 1/16 + ... = 1.
So you get the frog jumps x/2 + x/4 + x/8 + x/16 + ... = x units of distance in t/2 + t/4 + t/8 + t/16 + ... units of time.
That is, it jumps x units of distance in t units of time.
02 Dec 2017, 1:49 am
