"In mathematics, a power series (in one variable) is an infinite series of the form..."
A power series is a series that can be represented as an infinite polynomial.
Sequences: An ordered list of elements.
Limits and Convergence: The concept of a limit, and how it applies to sequences.
Series: A sum of an infinite sequence.
Tests for Convergence: Methods to determine if a series converges or diverges.
Alternating Series: A series in which terms alternate in sign.
Absolute and Conditional Convergence: Properties of convergent series.
Power Series: A series in which the terms contain variables raised to a power.
Taylor Series: A type of power series that approximates a function near a specific point.
Maclaurin Series: A type of Taylor series derived by centering at x=0.
Radius and Interval of Convergence: Properties of a power series that determine its convergence.
Differentiation and Integration of Power Series: Operations that can be applied to power series.
Applications of Power Series: How power series are used in calculus and other fields.
Geometric Series: A power series with a constant coefficient and a variable exponent raised to a power of a constant value. It has a ratio of the consecutive terms that remains constant.
Arithmetic Series: A power series in which the consecutive terms are equal to the sum of the previous term and a constant value.
Binomial Series: A power series with a binomial coefficient in the terms of the series.
Exponential Series: A power series with an exponential function as the variable factor.
Logarithmic Series: A power series with a logarithmic function involved either as the variable factor or the coefficient factor.
Trigonometric Series: A power series with a function involving the trigonometric functions, i.e., sin and cos functions.
Fourier Series: A power series that decomposes a function on a periodic interval into an infinite sum of sine and cosine functions, with the coefficients denoting the amplitudes of respective functions.
Legendre Series: A power series that utilizes the Legendre polynomials to define the function over a specified interval, often used in physics and engineering.
Hermite Series: A power series that employs the Hermite polynomials in the definition of the function, often used in quantum mechanics.
Bessel Series: A power series that employs the Bessel functions as the variable factor, often used to solve partial differential equations.
"an represents the coefficient of the nth term..."
"Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions."
"In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function."
"In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series."
"In such cases, the power series takes the simpler form..."
"Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions..."
"...generating functions (a kind of formal power series)..."
"...and in electronic engineering (under the name of the Z-transform)."
"The familiar decimal notation for real numbers can also be viewed as an example of a power series..."
"...with integer coefficients..."
"...but with the argument x fixed at 1⁄10."
"In number theory, the concept of p-adic numbers is also closely related to that of a power series."
"Borel's theorem implies that every power series is the Taylor series of some smooth function."
"When considering a Maclaurin series, the power series takes the simpler form..."
"Power series also occur in combinatorics as generating functions..."
"...and in electronic engineering (under the name of the Z-transform)."
"Borel's theorem implies that every power series is the Taylor series of some smooth function."
"In number theory, the concept of p-adic numbers is also closely related to that of a power series."
"Power series arise as Taylor series of infinitely differentiable functions."