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A power seriesPower series (in one variable) is an infinite seriesSeries of the form \(\mathop \sum \nolimits_{n = 0}^{\infty } c_{n} (x - x_{0})^{n}\) , where \(c_{n}\) are coefficients, \(x\) is a variable, and \(x_{0}\) is a constant called the center of the seriesSeries. These seriesSeries provide a powerful tool for approximating functions, solving differential equationsDifferential equations, and understanding the behavior of complex functions. By representing functions as power seriesPower series, we can often simplify calculations, analyze their properties more easily, and even extend their domains of definition. In this chapter, we explore Taylor and Maclaurin series, fundamental power series that provide a bridge between transcendental functions and simple polynomial approximations.

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Power Series Expansions (Taylor and Maclaurin Series)

  • 章节

  • 发表日期：

    2026年

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  • Farzin Asadi

摘要

A power seriesPower series (in one variable) is an infinite seriesSeries of the form \(\mathop \sum \nolimits_{n = 0}^{\infty } c_{n} (x - x_{0})^{n}\) , where \(c_{n}\) are coefficients, \(x\) is a variable, and \(x_{0}\) is a constant called the center of the seriesSeries. These seriesSeries provide a powerful tool for approximating functions, solving differential equationsDifferential equations, and understanding the behavior of complex functions. By representing functions as power seriesPower series, we can often simplify calculations, analyze their properties more easily, and even extend their domains of definition. In this chapter, we explore Taylor and Maclaurin series, fundamental power series that provide a bridge between transcendental functions and simple polynomial approximations.