- Solutions 1, S1 Bkgd
- Solutions 2, S2 Bkgd
- Problems 3
- The Discrete Fourier Transform
This is some reference material on various limits.
Choose . For we have and .
Now for any :
i.e. for . Now raise everything to the power and divide by where .
Now setting we have
for all . Let so that .
so as .
Here of course we are really combining the idea that and is unbounded so it's ok to let go to infinity as goes to infinity.
We can put this into a different form involving an arbitrary number between 0 and 1 by taking a special case where for some so as above . Then
In words any constant between zero and one raised to the y power goes to zero harder than y (raised to any power) goes to infinity, so their product in that limit goes to zero. In poker terminology: Exponentials beat powers which in turn beat straights.