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Monotonic Sequences and Examples https://youtu.be/vtg_3QJQJL0 - summary - playlist - notes In this video I go over the definition of increasing and decreasing sequences, which are thus denoted as Monotonic. #math #sequences #calculus #education #monotonicity
Using Mathematical Induction to find the Limit of a Recursive Sequence https://youtu.be/BHnjY22ddOM - summary - playlist - notes

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
MES Links
Bounded Sequences, Completeness Axiom, and the Monotonic Sequence Theorem https://youtu.be/8KIOeU07QEI - summary - playlist - notes In this video I first go over the definition of bounded sequences, then discuss the completeness axiom in number theory and…
Exercise 1: Limit of a Sequence and the Golden Ratio (Minus 1) https://youtu.be/tWKBS_9X5jE - summary - playlist - notes

In this video I show that the limit of a sequence is the same if the sequence was shifted by 1 term. I use this fact to solve for the limit of a recursive sequence by plugging in the limit into the recursion relation itself.

#math #sequences #goldenratio #education #calculus
MES Links
Exercise 1: Limit of a Sequence and the Golden Ratio (Minus 1) https://youtu.be/tWKBS_9X5jE - summary - playlist - notes In this video I show that the limit of a sequence is the same if the sequence was shifted by 1 term. I use this fact to solve for the…
Exercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence https://youtu.be/UKhcLVTQgl8 - summary - playlist - notes

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
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Exercise 3: Convergent Sequence inside a Continuous Function https://youtu.be/vdQa-zAA6w0 - summary - playlist - notes In this video I prove Theorem 3, which states that a convergent sequence with limit L can be plugged directly into a continuous function…
Exercise 4: Fibonacci's Original Rabbit Reproduction Sequence (and the Golden Ratio) https://youtu.be/NNNNKwGIKgI - summary - playlist - notes

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