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