Name: maclaurin series for cosine motion
Uploaded: 2026-06-08T09:38:39.681Z
Duration: 1 min 29 s
Channel: JoVE

泰勒级数将函数表示为在选定中心点附近收敛区间内的无穷幂级数。当中心为零时，该级数称为麦克劳林级数，其形式更为简单，使用 x 的幂次及在零处的导数。一个实际的例子是理想弹簧-质量系统做简谐运动。质量的运动可以用余弦函数来描述。在高速计算中，重复计算余弦函数可能带来较大的计算开销。为简化这一过程，对于接近零的小时间值，可使用余弦函数的麦克劳林级数进行近似。第一项给出一个等于初始位置的常数值。加入下一项后引入了曲率，使得近似结果与实际运动更加吻合。包含更高阶的项可提高精度，并更准确地反映振动的更多特征。在接近零的微小时间区间内，仅需少数几项即可提供良好的近似。随着时间远离中心点，需要更多的项来保持精度。这就是泰勒级数有价值的原因：它在简化复杂函数的同时，能够准确描述弹簧在中心附近区域的运动情况。

泰勒级数的应用

Applications of Taylor Series

泰勒级数提供了一种系统的方法，可将一个光滑函数表示为以选定点为中心的无穷多项式。当该中心点为零时，这种展开称为麦克劳林级数。这种形式特别有用，因为所有导数都在零处取值，通常使得系数更易于计算。在应用数学和物理学中，这类级数非常有价值，因为它们用多项式替代了复杂的函数，而多项式更容易进行分析和数值计算。一个标准的例子是余弦函数，其麦克劳林级数为\begin{equation*}\cos x = 1 - \jfrac{x^2}{2!} + \jfrac{x^4}{4!} - \jfrac{x^6}{6!} + \cdots\end{equation*}该展开式表明，余弦函数在零点附近可以用有限项的多项式进行近似。仅第一项可提供一个常数近似；加入二次项后，便引入了在余弦函数极大值点附近的向下曲率特征；增加更多的偶次幂项可进一步改善近似效果，使多项式能够在更大区间内更紧密地跟随余弦函数的振荡形态。这一概念可直接应用于处于简谐运动的理想弹簧-质量系统。若质量块从其最大位移 A 处开始运动，则其位置为\begin{equation*}x(t) = A\cos(\omega t)\end{equation*}其中 A 为振幅，⍵ 为角频率。利用余弦函数的麦克劳林级数可得\begin{equation*}x(t) \approx A\liparens{1 - \jfrac{(\omega t)^2}{2!} + \jfrac{(\omega t)^4}{4!} - \cdots}\end{equation*}对于非常小的数值 t，只需前几项即可获得良好的近似效果。这在高速计算中非常有用，因为重复计算三角函数的代价可能高于计算多项式。通过这种方式，泰勒级数在简化选定中心附近计算的同时，保留了弹簧运动的局部行为。

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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

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! + x^4/4! - x^6/6! + ... .

This expansion shows how cosine can be approximated near zero with only a few terms. The first term gives a constant approximation. Adding the quadratic term adds the downward curvature seen near the top of the cosine curve. More even-power terms improve the fit and help the polynomial follow the wave shape over a larger interval.

The same idea works for an ideal mass-spring system in simple harmonic motion. If the mass starts at its maximum displacement A, its position is x(t) = A cos(ωt), where A is the amplitude and ω is the angular frequency. Using the Maclaurin series for cosine gives x(t) ≈ A(1 - (ωt)^2/2! + (ωt)^4/4! - ...).

For very small values of t, only the first few terms are needed to get a good approximation. That makes the polynomial form useful in high-speed computing, where evaluating trigonometric functions may take more work than evaluating a polynomial. In this way, the Taylor series keeps the local motion of the spring while making the calculation simpler near the chosen center.

Maclaurin Series for Cosine Motion

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 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...

applications of taylor series

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 a Maclaurin series approximates cosine and simple harmonic motion with polynomials that are easier to compute near zero.

Maclaurin Series for Cosine Motion | 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

Maclaurin Series for Cosine Motion

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