The Precise Definition of a Limit of a Sequence https://youtu.be/baF4w9MeCiM - summary - playlist - notes

In this video I go over a more precise definition of the limit of a sequence by using the same approach as for the limit of a function. If the limit of a sequence exists, then there must be a positive number ε such that it has a corresponding number N where if the sequence term n is greater than N, then the absolute value of the difference between the n-th term of the sequence and the limit L is less than ε.

#math #science #education #algebra #sequences

Infinite Limits of Sequences at Infinity https://youtu.be/a4M4w3PhCd8 - summary - playlist - notes

In this video I go over the notation and precise definition for sequences that diverge to infinity when the number of terms go to infinity.

#math #science #education #algebra #sequences

Limit Laws for Sequences https://youtu.be/vgAitTQU2Fc - summary - playlist - notes

In this video I list the main limit laws for sequences, and which are just the general limit laws since they hold for convergent sequences as well.

#math #science #education #limits #sequences

Squeeze Theorem for Sequences https://youtu.be/fmTDMMUL31Q - summary - playlist - notes

In this video I rewrite the general squeeze theorem into its equivalent form for sequences. Essentially, if a sequence is always between two other sequences after a given set of terms, and those two sequences have the same limit, then our given sequence also has that same limit, and is said it gets "squeezed" between the two sequences.

#math #science #education #limits #sequences

Examples on Limits of Sequences https://youtu.be/80GBMrzuXVo - summary - playlist - notes

In this video I go over 4 examples on the limits of sequences and on determining if the limits exists or the sequence diverges.

#math #science #education #limits #sequences

Plugging a Sequence into a Function and Examples https://youtu.be/5Vc8QX_I7KA - summary - playlist - notes

In this video I go over Theorem 3, which states that if a sequence has a limit, then plugging it into a continuous function at that limit is the same as plugging in the limit.

#math #science #education #limits #sequences

Creating Graphs of Sequences https://youtu.be/AEubaxYkfX4 - summary - playlist - notes

In this video I discuss how to graph sequences manually as well as using Desmos online graphing calculator.

#math #science #Desmos #limits #sequences

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

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 how it is used to proof the monotonic sequence theorem.

#math #sequences #calculus #completeness #logic

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

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

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 at L, thus obtaining f(L).

#math #sequences #limits #education #calculus

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

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

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

Definition and Notation of Infinite Series - 3Speak - YouTube - Summary - Notes - Sections playlist - Sequences and Series playlist

In this video, I go over infinite series (or just series), which is defined as the summation of the terms of an infinite sequence of numbers. If we add an infinite sequence of numbers whose terms are each greater than 1, then the sum goes to infinity. But if the terms are fractions, then in some cases the sum tends towards a finite limit. In this case, we obtain a convergent series — otherwise, if there is no finite sum, then the series is called divergent.

#math #calculus #series #sequences #education

Theorem 2: Series of Sums is equal to Sum of Series - Hive - YouTube - Notes - Playlist - Sequences and Series

In this video, I go over Theorem 2, which states that we can move a constant out of the sum of a series as well as add or subtract series individually. These follow from the Limit Laws of Sequences, and I use this to prove that the series involving an addition of terms is equal to an addition of the individual series.

#math #calculus #series #sequences #education