Name: taylor series approximation and error
Uploaded: 2026-06-08T09:38:12.113Z
Duration: 1 min 30 s
Channel: JoVE

许多科学问题涉及难以直接计算的函数，例如交流电路中的振荡电压。在实际应用中，这些函数通常被更容易计算的多项式表达式所替代。泰勒定理提供了一种系统的方法来构建这些多项式逼近。该定理指出，一个函数在选定的点 a 附近等于一个 n 阶泰勒多项式加上一个余项。&#xA0;泰勒多项式是泰勒级数的有限部分，由函数在点 a 处的值及其直到 n 阶的导数组成。每一新增项都更精确地捕捉了函数的局部行为，因此在 a 附近，近似效果通常会提高。余项用于衡量近似误差，即同一位置处实际函数值与多项式值之间的差值。该误差取决于介于 a 与 x 之间某一点上 n 阶导数加一阶的导数。随着 n 增大，a 附近的误差通常会减小。如果对于区间 I 内所有的 x，当 n 趋近于无穷大时余项趋近于零，则该泰勒级数在该区间上收敛于函数。这意味着该级数在该区间上表示该函数。

泰勒级数的收敛性

Convergence of Taylor Series

泰勒级数提供了一种系统的方法，用一个在选定点附近与原函数高度吻合的多项式来逼近光滑函数。这种方法在科学与工程领域尤为重要，因为在这些领域中，某些函数可能难以直接计算，例如交流电路（AC）中的振荡电压。用多项式表达式替代复杂函数可以在保留关键局部行为的同时简化计算过程。泰勒定理为构建此类近似提供了理论基础，并可用于确定其逼近精度。阶数为 n 的泰勒多项式通过在中心点（通常记为 a）处匹配函数的值及其直到 n 阶的导数而构成。每一阶导数都捕捉了函数的额外局部信息，包括其变化率及更高阶的变化特征。通过逐次引入更高阶的导数，该多项式在点 a 的邻域内越来越接近原函数。这种系统性地引入高阶项的方式，能够优化近似效果并提高其局部精度。泰勒定理还引入了余项，用于量化函数与其 n 次泰勒多项式之间的差异。该余项取决于在 a 与所关注点之间的某个点上求值的 n + 1 阶导数。只要函数足够光滑，当 n 增大时，该余项在 a 附近的量值通常会减小。如果对于区间 I 内的所有值，当 n 趋近于无穷大时余项趋近于零，则泰勒级数在该区间上收敛于原函数。这种收敛性表明，无穷级数表示在整个收敛区间内都能准确再现原函数，从而确保其在理论上的有效性以及在实际应用中的可靠性。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Sequences and Series

数列与级数

KEY TERMS AND CONCEPTS

关键术语与概念

泰勒级数收敛

taylor series approximation and error

Taylor series use polynomials to approximate smooth functions near a chosen point. This makes hard-to-calculate functions easier to work with in science and engineering. A common example is an oscillatory voltage in an alternating current, or AC, circuit.

Taylor’s Theorem gives the rule for building these approximations and checking how accurate they are. A Taylor polynomial of order n matches a function’s value and its derivatives up to order n at a central point a. Each derivative adds more local information, such as the slope and higher changes in the curve.

As more derivative terms are included, the polynomial becomes a better local match to the original function near a. Taylor’s Theorem also includes a remainder term. This remainder measures the difference between the function and the n th-degree Taylor polynomial, and it depends on the derivative of order n plus one at some point between a and the point being studied.

When n gets larger, this remainder usually becomes smaller near a if the function is smooth enough. If the remainder goes to zero as n approaches infinity for every value in an interval I, then the Taylor series converges to the original function on that interval. This convergence means the infinite series matches the function throughout its interval of convergence and gives a reliable representation for use in calculations.

Taylor Series Approximation and Error

introduction to sequences

Zeno’s Paradox and Geometric Sequences

Zeno’s paradox shows how geometric sequences can describe motion that keeps getting smaller. In the example, a man walks toward a door but only covers half of the remaining distance each time. That creates a sequence of steps: one-half, one-quarter, one-eighth, and so on. This pattern is a geometric sequence because each term is half of the previous term. It can be written as a_n = 1/2^n, where n is the step number starting at 1. The terms shrink in a regular way, even though the sequence...

convergence of sequences

Bounded Sequences and the Limit Test

A sequence is an ordered list of numbers created one term at a time. In math analysis, students often ask whether the terms get closer and closer to a single real number as the index grows. When that happens, the sequence converges, and the number it approaches is called the limit. A graph can make this idea easier to see. As the sequence continues, its plotted points move toward a horizontal line. A simple example is the sequence of reciprocals of the natural numbers. Its terms get smaller as...

introduction to infinite series

Convergence and Divergence of Infinite Series

Infinite series are sums that continue without end, and their behavior depends on the pattern of their partial sums. A partial sum is the running total made by adding the first few terms of the series. If these running totals move toward one fixed number, the series converges. If they do not settle near a finite value, the series diverges. A water tank example shows how convergence can happen through repeated halving. Suppose a smart valve releases half of the remaining water every minute. In...

partial sums and series convergence

Bouncing Ball Partial Sums in Series

Partial sums help show how an infinite series can approach a finite total. An infinite series is made by adding the terms of an infinite sequence. Even though the adding never stops, some series still settle near a fixed value. This idea can model real processes where each step becomes smaller, such as a bouncing ball. A ball dropped from 1 meter rises to half its previous height after each bounce. After the first bounce, it reaches 0.5 meters. After the next bounce, it reaches 0.25 meters.

the integral test

Testing Series with Area Under Curves

Testing series with area under curves helps determine whether an infinite series converges or diverges. The Integral Test compares a series to an improper integral. It is useful when the terms are hard to add directly but match a related continuous function. The idea is to replace many discrete terms with the area under a curve. A glow stick gives a clear example. After it is activated, it starts bright and then slowly loses intensity over time. If the energy released each hour is recorded,...

comparison tests

Testing Series with Known Behavior

Comparison tests help determine whether an infinite series of positive terms converges or diverges. Instead of finding the sum directly, these methods compare the series with another series whose behavior is already known. That comparison gives a clear answer about whether the total stays finite or grows without bound. The direct comparison test works when each term of an unknown positive series can be matched to a second series term by term. If the unknown terms are less than or equal to the...

alternating series and absolute convergence

alternating-series-and-absolute-convergence

Damped Springs and Alternating Series

Damped spring motion can be modeled with alternating series. A mass on a vertical spring moves above and below a central equilibrium point. In an ideal spring, the motion would keep the same amplitude. In a damped spring, resistive forces such as air resistance or internal friction make each swing smaller over time. This motion is often written as a sinusoidal function with an exponential decay factor. The sine term describes the repeating back-and-forth motion. The decay factor lowers the...

the ratio test

Compound Interest and Series Growth

Compound interest grows an investment by adding interest to both the original amount and the interest already earned. With a 4% annual compound interest rate, each year’s balance becomes 1.04 times the previous year’s balance. That pattern creates a geometric sequence, with 1.04 as the common ratio or repeated multiplier. Mathematicians use the Ratio Test to check whether an infinite series has a finite sum or keeps growing without limit. The test compares consecutive terms in the series and...

power series and their properties

Power Series as Function Approximations

Power series can represent a function as an infinite sum of terms with powers of a variable. The terms are usually written around a central value, called the center. That center is shown as a, so the powers become (x − a)n, where n is a non-negative integer. The coefficients in the series control the behavior of the function. When the center is zero, the series becomes a power expansion in x. It then looks much like an infinite polynomial. This form can model complex functions with high...

interval and radius of convergence

Testing Power Series Convergence Bounds

Power series converge only for certain input values, so it is important to find the range where they work. The radius of convergence gives the distance from the center where the series still converges. The interval of convergence shows the real x-values that fit inside that range. A general power series can be written as a sum of terms of the form c_n(x - a)^n, where a is the center. To find the radius of convergence, the ratio test is often used. It compares consecutive terms by taking the...

bessel function of order zero

Modeling Ripple Decay with Bessel Functions

Bessel functions help model ripple decay in waves with radial symmetry. A simple example is the circular wave that spreads outward after a stone drops into a still pond. The disturbance starts at the center and moves away in all directions. As the wavefront grows, the same energy spreads over a larger circumference. That causes the wave amplitude, or height, to decrease with distance from the center. Simple sine and cosine functions do not describe this change well because their amplitudes...

real-world applications of power series

Pendulum Motion and the Small-Angle Approximation

Pendulum motion can be simplified with the small-angle approximation and a power series. A simple pendulum follows Newton’s Second Law in rotational form, which links the net torque on the bob to its angular acceleration. That relationship leads to a second-order differential equation. The equation is nonlinear because the angular acceleration depends on the sine of the displacement angle. To make the model easier to solve, the sine function is expanded with a Maclaurin series. A Maclaurin...

taylor series

Matching Functions with Taylor Coefficients

Taylor series match a smooth function near a chosen center by using its derivatives. The center is the point where the function is analyzed, and the goal is to build a power series that follows the function’s local behavior. In this expansion, each coefficient is chosen so the series and the function have the same value and the same derivatives at the center. Start with a power series centered at x = a. It has the form Σn=0∞ cn(x − a)n. When x = a, every term with (x − a) becomes zero, so only...

convergence of taylor series

Taylor series use polynomials to approximate smooth functions near a chosen point. This makes hard-to-calculate functions easier to work with in science and engineering. A common example is an oscillatory voltage in an alternating current, or AC, circuit. Taylor’s Theorem gives the rule for building these approximations and checking how accurate they are. A Taylor polynomial of order n matches a function’s value and its derivatives up to order n at a central point a. Each derivative adds more...

applications of taylor series

Maclaurin Series for Cosine Motion

Maclaurin series turn a smooth function into an infinite polynomial centered at zero. A Maclaurin series is a Taylor series with the center at zero, so every derivative is taken at that point. That setup often makes the coefficients easier to find. These series are useful in applied mathematics and physics because they replace complicated functions with polynomials that are easier to analyze and compute. One standard example is the cosine function. Its Maclaurin series is cos x = 1 - x^2/2! +...

binomial series

Power Series for Fractional Exponents

The binomial series shows how to expand powers like (1 + x)^m into a power series. It extends the binomial theorem, which works for positive integer exponents. The series also applies to fractional and negative exponents, so it reaches beyond the finite case. It comes from the Maclaurin series for f(x) = (1 + x)^m, where m is any real number. The expansion begins as 1 + mx + m(m - 1)x^2/2! + m(m - 1)(m - 2)x^3/3! + ... . In compact form, it is written as (1 + x)^m = sum from k = 0 to infinity...

key terms and concepts

Learn how Taylor polynomials approximate smooth functions, how the remainder measures error, and when a Taylor series converges to the original function.

Taylor Series Approximation and Error | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Taylor Series Approximation and Error

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