power series
Noun: 1. A mathematical expression: A power series is an infinite series of terms. Each term consists of a constant coefficient multiplied by a variable raised to a successively increasing non-negative integer power (e.g., x⁰, x¹, x², x³...). It is a fundamental tool in calculus and analysis for representing functions.
The term "power series" is used to describe a specific type of mathematical series. It is typically followed by a phrase specifying the variable and the center of the series. * It is often used in the context of convergence, where one discusses the interval or radius within which the series converges to a function. * Common verbs used with it include: expand a function into a power series, find the power series for a function, determine the convergence of a power series.
- The power series expansion of the exponential function eˣ is 1 + x + x²/2! + x³/3! + ...
- A Taylor series is a specific type of power series.
- The function 1/(1-x) can be represented by the geometric power series Σ xⁿ for |x| < 1.
- Analyzing the radius of convergence is crucial when working with a power series.
- Formal power series: In algebra, a power series may be treated without concern for convergence, focusing solely on its algebraic properties.
- Power series solution: A method for solving differential equations by assuming the solution can be written as a power series and solving for the coefficients.
- Taylor series (n.): A power series that represents a function and is constructed from the function's derivatives at a single point.
- Maclaurin series (n.): A Taylor series centered at zero.
- Series expansion (n.): The general expression of a function as an infinite sum of terms, which may be a power series.
- Coefficient (n.): The constant factor multiplying the variable in each term of a power series.
- Infinite series (in a general mathematical context, though not all infinite series are power series).
- Radius of convergence: A non-negative real number that represents the distance from the center within which the power series converges.
- Interval of convergence: The set of all real numbers for which the power series converges.
- Analytic function: A function that can be locally expressed as a convergent power series.
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the sum of terms containing successively higher integral powers of a variable
