Name: power series as function approximations
Uploaded: 2026-06-08T09:38:23.494Z
Duration: 1 min 30 s
Channel: JoVE

A power series is an infinite sum that represents a function within its interval of convergence. The series is defined by coefficients multiplied by powers of x minus a, where a is the center of the series. When a equals zero, the terms are simple powers of x.This structure makes a power series act like an infinite polynomial.For example, setting all coefficients to one gives the geometric series with ratio x.This series converges to 1 divided by 1 minus x only when the absolute value of x is less than 1. This defines the interval where the series behaves like the function.Graphs of the series&#x27; partial sums, formed by adding the first few terms, closely match the function near the center.As x moves farther from the center, adding more terms may improve accuracy. If the absolute value of x is greater than 1, the series diverges and does not represent the function.In AC circuits, trigonometric functions model voltage and current. Power series represent these functions as infinite sums of polynomial terms. This lets circuit simulation tools calculate accurate values using basic arithmetic.

power series and their properties

A power series represents a function as an infinite sum of terms involving powers of a variable, typically expressed relative to a central value. This center, denoted as a, shifts the powers to terms of (x &#x2212; a)n, where n is a non-negative integer and the coefficients define the behavior of the function. When the center is zero, the power series becomes a straightforward expansion in powers of x, closely resembling an infinite polynomial. This structure allows power series to model complex functions with high accuracy within specific regions.Convergence and ApproximationThe convergence of a power series depends on the value of x and the interval around the center. The geometric series&#x2014;a foundational example&#x2014;has all coefficients equal to one and converges to the function 1/(1&#x2212;x) when the absolute value of x is less than one. This defines the radius of convergence. Within this interval, partial sums of the series&#x2014;finite approximations using the first few terms&#x2014;provide increasingly accurate representations of the target function. Closer to the center, fewer terms are needed. However, as x approaches the boundary of convergence, additional terms are required to maintain precision. Beyond this boundary, the series diverges and ceases to accurately represent the function.Applications in Engineering and PhysicsPower series are especially valuable in applied contexts where exact expressions are difficult to manage. For instance, trigonometric functions can be expanded into power series and substituted with polynomial approximations in alternating current (AC) circuit models. This conversion simplifies the underlying mathematics, enabling easier computation of voltage and current values in simulations. By transforming periodic functions into polynomial forms, simulation tools can handle complex circuits more efficiently, making power series a practical tool in both theoretical and engineering applications.

Power Series and Their Properties

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Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Sequences and Series

수열과 급수

KEY TERMS AND CONCEPTS

주요 용어 및 개념

파워 시리즈와 그 속성

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 accuracy in a limited region around the center.

The convergence of a power series depends on the value of x and the interval around the center. A geometric series is a basic example. It has all coefficients equal to one and converges to 1/(1 − x) when the absolute value of x is less than one. That range sets the radius of convergence.

Inside the interval of convergence, partial sums give useful approximations. A partial sum is a finite version of the series that uses only the first few terms. These sums become more accurate as more terms are added. Near the center, fewer terms are needed. As x gets closer to the boundary of convergence, more terms are required to keep the same level of accuracy. Beyond that boundary, the series diverges and no longer matches the function well.

Power series are also useful in engineering and physics. Trigonometric functions can be expanded into power series and then replaced with polynomial approximations in alternating current, or AC, circuit models. This makes the math easier to handle when calculating voltage and current in simulations. By turning periodic functions into polynomial form, simulation tools can work with complex circuits more efficiently.

Power Series as Function Approximations

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 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 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 power series use centers, partial sums, and radius of convergence to approximate functions, with a geometric series example and AC circuit applications.

Power Series as Function Approximations | 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

Power Series as Function Approximations

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