Name: power series for fractional exponents
Uploaded: 2026-06-08T09:38:22.653Z
Duration: 1 min 30 s
Channel: JoVE

二项式级数是函数 1 加 x 的 m 次幂的麦克劳林级数展开式。当 m 不是正整数时，该级数是无穷级数，因为分子项永远不会变为零。当 x 的绝对值小于 1 时，该级数收敛。该级数可用来近似诸如狭义相对论中总能量之类的函数。对于一个静止质量为 m naught、以速度 v 运动的物体，其总能量取决于它的质量、速度以及光速 c。为了应用二项式级数，能量表达式被重写为 1 加上一个微小项的幂次形式。当速度低于 c 时，该项的量值保持在 1 以下，满足收敛条件。展开表达式后得到速度的各次幂项。当 v 远小于 c 时，每一后续项趋近于零。保留前两项可得到总能量的近似值。第一项表示物体的静止能量。从总能量中减去此项后，剩下第二项，即经典动能公式。这展示了二项式级数如何近似复杂表达式。

二项式级数

Binomial Series

二项式级数将熟悉的二项式定理从有限的多项式展开推广到无穷级数展开。这一区别至关重要：二项式定理适用于正整数指数，而二项式级数的适用范围更广，包括分数指数和负指数。该级数由 (1 + x)m 的麦克劳林级数得到，其中 m 为任意实数指数，且当 |x| &lt; 1 时，该展开式收敛。熟悉的二项式定理是\begin{equation*}(y+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k y^{n-k}\end{equation*}令 y = 1 可得\begin{equation*}(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k\end{equation*}通过考虑 f(x) = (1 + x)m 可将此概念推广。利用麦克劳林级数，\begin{equation*}(1+x)^m=1+m x+\jfrac{m(m-1)}{2!}x^2+\frac{m(m-1)(m-2)}{3!}x^3+\cdots\end{equation*}紧凑形式下，\begin{equation*}(1+x)^m=\sum_{k=0}^{\infty}\binom{m}{k}x^k\end{equation*}其中\begin{equation*}\binom{m}{k}= \frac{m(m-1)(m-2)\cdots(m-k+1)}{k!}\end{equation*}对于正整数 m，其因数最终会包含零，因此该级数终止，从而成为二项式定理。对于负数或非整数的 m，因数不会终止，从而形成无穷二项式级数。

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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

二项级数

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 of binomial(m, k)x^k.

The generalized binomial coefficient is written as binomial(m, k) = m(m - 1)(m - 2)...(m - k + 1)/k!. For a positive integer m, these factors eventually include zero. The series then stops, and it becomes the usual binomial theorem.

For negative or noninteger m, the factors do not stop. That gives an infinite binomial series. The expansion converges for |x| &lt; 1.

Power Series for Fractional Exponents

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...

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introduction to infinite series

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partial sums and series convergence

Bouncing Ball Partial Sums in Series

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the integral test

Testing Series with Area Under Curves

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alternating series and absolute convergence

alternating-series-and-absolute-convergence

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the ratio test

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power series and their properties

Power Series as Function Approximations

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interval and radius of convergence

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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...

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real-world applications of power series

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taylor series

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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

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 the binomial series expands (1 + x)^m for real exponents, including fractional and negative powers, and when the series converges.

Power Series for Fractional Exponents | 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 for Fractional Exponents

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