Number theory | Topology | Integer sequences
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by and ; they differ only for n = 1, where and . For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials , with and . The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program. (Wikipedia).
Solve a Bernoulli Differential Equation (Part 2)
This video provides an example of how to solve an Bernoulli Differential Equation. The solution is verified graphically. Library: http://mathispower4u.com
From playlist Bernoulli Differential Equations
B25 Example problem solving for a Bernoulli equation
See how to solve a Bernoulli equation.
From playlist Differential Equations
Solve a Bernoulli Differential Equation (Part 1)
This video provides an example of how to solve an Bernoulli Differential Equation. The solution is verified graphically. Library: http://mathispower4u.com
From playlist Bernoulli Differential Equations
Solving the Bernoulli Differential Equation x^2(dy/dx) + y^2 = xy
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys How to solve a Bernoulli Differential Equation
From playlist Differential Equations
B24 Introduction to the Bernoulli Equation
The Bernoulli equation follows from a linear equation in standard form.
From playlist Differential Equations
Ex: Solve a Bernoulli Differential Equation Using Separation of Variables
This video explains how to solve a Bernoulli differential equation. http://mathispower4u.com
From playlist Bernoulli Differential Equations
Ex: Solve a Bernoulli Differential Equation Using an Integrating Factor
This video explains how to solve a Bernoulli differential equation. http://mathispower4u.com
From playlist Bernoulli Differential Equations
Solve a Bernoulli Differential Equation Initial Value Problem
This video provides an example of how to solve an Bernoulli Differential Equations Initial Value Problem. The solution is verified graphically. Library: http://mathispower4u.com
From playlist Bernoulli Differential Equations
Bernoulli Differential Equations: Differential Equations Lesson #4
Some of the links below are affiliate links. As an Amazon Associate I earn from qualifying purchases. If you purchase through these links, it won't cost you any additional cash, but it will help to support my channel. Thank you! ►PRODUCT RECOMMENDATIONS https://www.amazon.com/shop/brithem
From playlist Differential Equations
Advice | Variants of the Bernoulli numbers via an AI approach to maths research | Wild Egg Maths
We advocate a simple minded AI approach to pure maths research: start with a basic, central object in mathematics, and just systematically explore the possibilities adjacent to it, where adjacent means roughly that we perform small variants and see what happens. To illustrate the strategy
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Advice for research mathematicians | Bernoulli numbers and Faulhaber's sums of powers | WIld Egg
The Bernoulli numbers are an intriguing family of rational numbers that arise in many areas of analysis. We introduce them in the context of J. Faulhaber's formulas for sums of powers of natural numbers, which in fact give us another important family of polynomials or polynumbers. These n
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Advice for Amateur Mathematicians |Bernoulli Polynumbers and Euler Maclaurin summation | Wild Egg
We introduce the Bernoulli polynomials built from the Bernoulli numbers and binomial coefficients. These are important ingredients in a famous formula of Calculus that connects finite summations and integrations due to Euler and Maclaurin. More details about this particular application wil
From playlist Maxel inverses and orthogonal polynomials (non-Members)
MIT 6.041 Probabilistic Systems Analysis and Applied Probability, Fall 2010 View the complete course: http://ocw.mit.edu/6-041F10 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
The Bernoullis: When Math is the Family Business
If you’ve ever taken a science or math class, you’ve probably seen the name "Bernoulli" -- and maybe you assumed it was one person, but that family had a squad of mathematicians. Hosted by: Hank Green ---------- Support SciShow by becoming a patron on Patreon: https://www.patreon.com/scis
From playlist Uploads
Bernoulli Distribution Probability & PDF
Examples of finding probabilities with the Bernoulli distribution PDF. Expected value and variance, independence and links to other distributions.
From playlist Probability Distributions
MIT 6.041 Probabilistic Systems Analysis and Applied Probability, Fall 2010 View the complete course: http://ocw.mit.edu/6-041F10 Instructor: John Tsitsiklis License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
The Basel Problem Part 1: Euler-Maclaurin Approximation
This is the first video in a two part series explaining how Euler discovered that the sum of the reciprocals of the square numbers is π^2/6, leading him to define the zeta function, and how Riemann discovered the surprising connection between the zeroes of the zeta function and the distrib
From playlist Analytic Number Theory
Faulhaber's Formula and Bernoulli Numbers | Algebraic Calculus One | Wild Egg
This is a lecture in the Algebraic Calculus One course, which will present an exciting new approach to calculus, sticking with rational numbers and high school algebra, and avoiding all "infinite processes", "real numbers" and other modern fantasies. The course will be carefully framed on
From playlist Algebraic Calculus One from Wild Egg
Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula)
The longest Mathologer video ever! 50 minutes, will this work? Let's see before I get really serious about that Kurosawa length Galois theory video :) Today's video is another self-contained story of mathematical discovery covering millennia of math, starting from pretty much nothing and
From playlist Recent videos
How to Solve a Bernoulli Differential Equation
How to Solve a Bernoulli Differential Equation
From playlist Differential Equations