Binet's formula for the Fibonacci numbers. Infinite sums and generating functions to prove it!

Описание: Binet's formula is an explicit formula for the Fibonacci numbers that involves the golden ratio. In this video, I use generating functions and infinite sums to derive Binet's formula. As a bonus fact, we learn how to convert between miles and kilometers with the Fibonacci numbers. The video goes over the notes posted at https://sites.uoguelph.ca/nicam/files...

KNOWN ERRORS:
-The formula for S right below "Fact 3" has a minus sign mistake in the video (the current version of the notes has it correct)
-On the last slide I wrote "+" in Binet's formula instead of minus

0:00 What is this video
0:45 What are the Fibonacci numbers
1:30 What is Binet's formula
2:10 Using infinite sums
3:05 Exercise 0
6:20 Exercise 1
8:27 Technicalities I won't think about
9:18 Exercise 2
10:38 Exercise 3
13:55 What are generating functions
14:25 Example 1
15:42 Example 2
16:17 Example 3
16:50 Why the generating function for the Fibonacci numbers?
17:45 Working out Fact 1 attempt 1
22:40 Working out Fact 1 attempt 2
26:25 Fact 1
27:15 Phi and Beta
28:35 Fact 2 Factoring with phi and beta
30:50 Fact 3 Partial Fraction Trick
33:20 Fact 3 Replacing with an infinite sum
34:14 Putting it all together
36:46 Final answer and recap
38:10 Miles to km