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

The binomial series is the Maclaurin series expansion for the function one plus x raised to the power m. The series is infinite when m is not a positive integer because the numerator terms never become zero. It converges when the absolute value of x is less than one.This series can approximate functions like total energy in special relativity. For an object with rest mass m naught moving at velocity v, total energy depends on its mass, velocity, and the speed of light, c.To apply the binomial series, the energy expression is rewritten as one plus a small term raised to a power. At velocities slower than c, the magnitude of this term stays below one, satisfying the condition for convergence.Expanding the expression gives terms with increasing powers of velocity. When v is much less than c, each subsequent term approaches zero. Retaining the first two terms gives an approximation for the total energy.The first term represents the object&#x27;s rest energy. Subtracting this from the total energy leaves the second term, which is the classical formula for kinetic energy. This shows how the binomial series approximates complex expressions.

binomial series

Binomial Series

The binomial series extends the familiar binomial theorem from finite polynomial expansions to infinite series expansions. This distinction is important: the binomial theorem applies to positive integer exponents, while the binomial series applies more broadly, including fractional and negative exponents. It is obtained from the Maclaurin series of (1 + x)m, where m is any real exponent, and the expansion converges for |x| &lt; 1.The familiar binomial theorem is\begin{equation*}(y+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k y^{n-k}\end{equation*}Setting y = 1 gives\begin{equation*}(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k\end{equation*}This idea is generalized by considering f(x) = (1 + x)m. Using the Maclaurin series,\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*}In compact form,\begin{equation*}(1+x)^m=\sum_{k=0}^{\infty}\binom{m}{k}x^k\end{equation*}where\begin{equation*}\binom{m}{k}= \frac{m(m-1)(m-2)\cdots(m-k+1)}{k!}\end{equation*}For a positive integer m, the factors eventually include zero, so the series terminates and becomes the binomial theorem. For negative or noninteger m, the factors do not terminate, giving an infinite binomial series.

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Calculus

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Sequences and Series

KEY TERMS AND CONCEPTS

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

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taylor series

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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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