Mathematical series | Calculus
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise. In modern terminology, any (ordered) infinite sequence of terms (that is, numbers, functions, or anything that can be added) defines a series, which is the operation of adding the ai one after the other. To emphasize that there are an infinite number of terms, a series may be called an infinite series. Such a series is represented (or denoted) by an expression like or, using the summation sign, The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the series. That is, When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent. The notation denotes both the series—that is the implicit process of adding the terms one after the other indefinitely—and, if the series is convergent, the sum of the series—the result of the process. This is a generalization of the similar convention of denoting by both the addition—the process of adding—and its result—the sum of a and b. Generally, the terms of a series come from a ring, often the field of the real numbers or the field of the complex numbers. In this case, the set of all series is itself a ring (and even an associative algebra), in which the addition consists of adding the series term by term, and the multiplication is the Cauchy product. (Wikipedia).
Introductory talk on series. Defining a series as a sequence of partial sums.
From playlist Advanced Calculus / Multivariable Calculus
Calculus 2: Infinite Sequences and Series (7 of 62) What is a Series?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what are series; and differences between arithmetic series and its common difference, and geometric series and its common ratio. Next video in the series can be seen at: https://youtu.be/3oAG
From playlist CALCULUS 2 CH 14 SERIES AND SEQUENCES
Definition and examples of series. In this video, I define the notion of a series using partial sums, and give a couple of examples of series. I also show the fact used in many times in calculus that a series converges if and only if it is bounded, which is used many times in calculus. Enj
From playlist Series
What is the formula to find the sum of an arithmetic sequence
👉 Learn all about series. A series is the sum of the terms of a sequence. Just like in sequences, there are many types of series, among which are: arithmetic and geometric series. An arithmetic series is the sum of the terms of an arithmetic sequence. A geometric series is the sum of the t
From playlist Series
This video introduces geometric series. http://mathispower4u.yolasite.com/
From playlist Series (Algebra)
Free ebook http://bookboon.com/en/learn-calculus-2-on-your-mobile-device-ebook What is a telescoping series and how do we calculate its value? A very important and useful concept in math.
From playlist A second course in university calculus.
Finding the sum or an arithmetic series using summation notation
👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is
From playlist Series
Introduction to Arithmetic Series
This video introduces series and the various formulas used when working with arithmetic series.
From playlist Series (Algebra)
(New Version Available) Arithmetic Series
New Version: https://youtu.be/GZH68SubgRE This video introduces arithmetic series. http://mathispower4u.yolasite.com/
From playlist Sequences and Series
The finite part of infinity (ONLINE) by Joseph Samuel
Vigyan Adda The finite part of infinity (ONLINE) Speaker: Joseph Samuel (RRI & ICTS-TIFR, Bengaluru) When: 4:30 pm to 6:00 pm Sunday, 24 October 2021 Where: Livestream via the ICTS YouTube channel Abstract: - Ramanujan's notebooks contain the equation 1+2+3....= - 1/12. While this see
From playlist Vigyan Adda
The man who invented calculus: the life and work of Madhava (Lecture 1) by P P Divakaran
MADHAVA LECTURES THE MAN WHO INVENTED CALCULUS: THE LIFE AND WORK OF MADHAVA SPEAKER: P P Divakaran (formerly Professor of Physics at TIFR-Mumbai) DATE: 11 February 2020, 16:00 VENUE: ICTS-TIFR, Bengaluru Lecture 1: Tuesday, February 11, 2020 at 16:00 Lecture 2: Thursday, February 13
From playlist Madhava Lectures
Mathematics of Series with the Symbolic Computing Package
To learn more about Wolfram Technology Conference, please visit: https://www.wolfram.com/events/technologyconference/ Speaker: Youngjoo Chung Wolfram developers and colleagues discussed the latest in innovative technologies for cloud computing, interactive deployment, mobile devices, and
From playlist Wolfram Technology Conference 2017
Greek Mathematics: The Beginning of Greek Math & Greek Numerals
Welcome to the History of Greek Mathematics mini-series! This series is a short introduction to Math History as a subject and the some of the important theorems created in ancient Greece. You are watching the first video in the series. If this series interested you check out our blog for
From playlist The History of Greek Mathematics: Math History
Featuring Professor Edward Frenkel. More links & stuff in full description below ↓↓↓ Okay, the links... 1. New vid on the Riemann Hypothesis explains where -1/12 comes from: http://youtu.be/d6c6uIyieoo 2.The original videos at http://youtu.be/w-I6XTVZXww AND http://youtu.be/E-d9mgo8FGk 3
From playlist Animations by Pete McPartlan
The man who invented calculus: the life and work of Madhava (Lecture 3) by P P Divakaran
MADHAVA LECTURES THE MAN WHO INVENTED CALCULUS: THE LIFE AND WORK OF MADHAVA SPEAKER: P P Divakaran (formerly Professor of Physics at TIFR-Mumbai) DATE: 11 February 2020, 16:00 VENUE: ICTS-TIFR, Bengaluru Lecture 1: Tuesday, February 11, 2020 at 16:00 Lecture 2: Thursday, February 13
From playlist Madhava Lectures
Orthonormal Bases Vs Fourier Series Part 2
Lecture with Ole Christensen. Kapitler: 00:00 - Proof Of Thrm 4.7.2 Continued; 11:00 - Connection To Fourier Series; 11:15 - L2(-Pi,Pi); 16:00 - Complex Fourier Series; 17:45 - Convergence?; 35:45 - Parseval Identity;
From playlist DTU: Mathematics 4 Real Analysis | CosmoLearning.org Math
Pure Mathematics Book with Solutions to All Problems(from 1960's England)
In this video I go over a book you have probably never heard of, I can't even find it on amazon! It is a book on Pure Mathematics written in the 1960s from London England. It covers a super wide range of topics and has solutions to every single problem in the book which is awesome. I talk
From playlist Book Reviews
History of Indian Mathematics Part V: Ramanujan's Discoveries
Learn about Srinivasa Ramanujan, one of history's greatest mathematical minds! Check out the whole series on the blog: https://centerofmathematics.blogspot.com/2019/11/history-of-indian-mathematics.html
From playlist History of Indian Mathematics
Advice for amateur mathematicians | More magic, with Euler numbers and Euler polynomials | Wild Egg
From the Bernoulli numbers and Bernoulli polynomials, it is a small step to consider Euler numbers and Euler polynomials. We of course pursue our basic strategy of incorporating these things into our two dimensional number theoretic point of view, and then utilizing linear algebra / matrix
From playlist Maxel inverses and orthogonal polynomials (non-Members)
The geometric series.
From playlist Advanced Calculus / Multivariable Calculus