sequences Tag Posts Index

Theorem 2: Series of Sums is equal to Sum of 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

Theorem 2: Series of Sums is equal to Sum of 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

Theorem 1 and the Test for the Divergence of Infinite Series

In this video, I go over Theorem 1, which states that the terms of a convergent series approach zero. Conversely, if the terms don't approach zero, the series diverges,

Telescoping Sum: All the terms of this series cancel out except for the first and last term

In this video, I go over an infinite series in which arises an example of the famous telescoping sum such that all the terms of the series cancel except for the

Geometric Series: Sum of x^n series = 1/(1 - x) if |x| is less than 1

In this video, I show that the infinite series xn = x0 + x1 + x2 + ... is just a geometric series, and converges if the absolute value of x is less than 1. By common

Converting the number 2.3171717... into the fraction 1147/495 via the Geometric Series

In this video, I go over an example of converting the number 2.3171717... (with infinite repeating 17) into a fraction by writing the repeating 17s as an infinite

Examples on Identifying Geometric Series and Determining if it Converges or Diverges

In this video, I go over two examples and two methods of identifying if a series is a geometric series, and then finding its sum if it converges. In the first example,

Geometric Series: Deriving the Sum using an Ingenious Method (and via Similar Triangles)

In this video, I go over the geometric series, which is the sum a + a r + a r^2 + ..., and show that if the absolute value of the common ratio r is less than 1,

Definition and Notation of Infinite Series

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

Definition and Notation of Infinite Series

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

History of the Fibonacci Sequence and the Golden Ratio

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

History of the Fibonacci Sequence and the Golden Ratio

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

Exercise 4: Fibonacci's Original Rabbit Reproduction Sequence (and the Golden Ratio)

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

Exercise 4: Fibonacci's Original Rabbit Reproduction Sequence (and the Golden Ratio)

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

Exercise 3: Convergent Sequence inside a Continuous Function

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).

Exercise 3: Convergent Sequence inside a Continuous Function

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).

Exercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence

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

Exercise 2: If the limit of the absolute values of a sequence is zero, so is the sequence

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

Exercise 1: Limit of a Sequence and the Golden Ratio (Minus 1)

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

Exercise 1: Limit of a Sequence and the Golden Ratio (Minus 1)

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