History of the Fibonacci Sequence and the Golden Ratio https://youtu.be/ube-OSVYMPY

In this video I go over the history of the Fibonacci sequence and Fibonacci numbers as well as their relation to the Golden Ratio and Golden angle, rectangle, and spiral.

#math #fibonacci #goldenratio #calculus #sequences

Exercise 4: Fibonacci's Original Rabbit Reproduction Sequence (and the Golden Ratio) https://youtu.be/NNNNKwGIKgI

In this video I go over the first appearance of the famous Fibonacci sequence and show that the limit of the ratio of two consecutive terms is equal to the Golden Ratio. The Italian mathematician Leonardo Bonacci, or more commonly known as Fibonacci (short for "filius Bonacci or "son of Bonacci"), first wrote the Fibonacci sequence in 1202 when analyzing the population growth of an idealized rabbit population. 

#math #sequences #fibonaccisequence #goldenratio #calculus

Exercise 3: Convergent Sequence inside a Continuous Function https://youtu.be/vdQa-zAA6w0

In this video I prove Theorem 3, which states that a convergent sequence with limit L can be plugged directly into a continuous function at L, thus obtaining f(L).

#math #sequences #limits #education #calculus

Exercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence https://youtu.be/UKhcLVTQgl8

In this video I prove Theorem 2, which states that if the limit of the absolute values of a sequence is zero, then the sequence itself approaches zero. I solved this using both Squeeze Theorem and the precise definition of a limit.

#math #sequences #absolutevalue #education #calculus

Bounded Sequences, Completeness Axiom, and the Monotonic Sequence Theorem https://youtu.be/8KIOeU07QEI

In this video I first go over the definition of bounded sequences, then discuss the completeness axiom in number theory and how it is used to proof the monotonic sequence theorem. 

#math #sequences #calculus #completeness #logic

Using Mathematical Induction to find the Limit of a Recursive Sequence https://youtu.be/BHnjY22ddOM

In this video I go over an example of a sequence that is defined by a recurrence relationship, and then use mathematical induction to prove it is increasing, bounded, thus convergent, and finally its actual limit.

#math #sequences #mathematicalinduction #calculus #education