Power Series Expansion | Taylor & Maclaurin's Infinite Series Explained | Engineering Essentials

In this Real Analysis video from Engineering Essentials, we dive deep into Power Series Expansions and explore Taylor and Maclaurin's Infinite Series. Learn how functions can be represented as infinite series, the role of Taylor and Maclaurin series in approximation, and their applications in solving mathematical problems. This video is a must-watch for bachelor's and master's students in Mathematics, Physics, Statistics, and Engineering. It is part of a comprehensive Real Analysis playlist covering Real Numbers, Sequences, Mean Value Theorems, and more.

Keywords: Power Series Expansion, Taylor Series, Maclaurin Series, Real Analysis, Infinite Series, Real Numbers, Convergence, Mathematics, Engineering Mathematics, Physics, Statistics

Hashtags: #PowerSeries #TaylorSeries #MaclaurinSeries #RealAnalysis #Mathematics #EngineeringMathematics #Physics #Statistics #InfiniteSeries

Short Technical Descriptions:
Power Series Expansion: A representation of a function as an infinite sum of terms involving powers of a variable, often used for approximating functions near a given point.
Taylor Series: A special case of power series expansion that represents a function as an infinite sum of derivatives evaluated at a specific point.
Maclaurin Series: A Taylor series centered at zero, offering a simpler form of function expansion for common functions like exponentials and trigonometric functions.


Power Series Expansion | Taylor & Maclaurin's Infinite Series #maths #eranand

In mathematics, a Taylor series or Taylor expansion is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
The Taylor series for a function f(x) centered at x=a is given by:
f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
where f'(a), f''(a), f'''(a), etc. are the first, second, third, etc. derivatives of f(x) evaluated at x=a.
The Taylor series can be used to represent a function in an infinite number of ways, each with a different degree of accuracy. The first few terms of the Taylor series often provide a good approximation of the function near the point x=a.
The Taylor series can be used to find the values of functions at points where they are not defined, to find derivatives of functions, to integrate functions, and to solve differential equations.
The Taylor series is a powerful tool that can be used to solve a variety of problems in mathematics and physics.

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