Binomial series

The expansion of \((1+x)^\alpha\) as a power series with generalized binomial coefficients. Its convergence depends on the exponent and the value of \(x\).

Generalized binomial coefficient: For an exponent \(\alpha\) and integer \(k\ge 0\), the product \(\alpha(\alpha-1)\cdots(\alpha-k+1)/k!\), with value 1 at \(k=0\). These coefficients generate every term of the series, including when the exponent is not an integer.
Binomial theorem: The finite expansion of \((a+b)^n\) for a nonnegative integer exponent \(n\). It is the finite integer-exponent case that the binomial series extends.
Newton's generalized binomial theorem: Isaac Newton's infinite-series extension of the binomial theorem to noninteger exponents. Newton's formulation established the expansion's use beyond finite integer powers.
Finite binomial expansion: The polynomial expansion of a binomial raised to a nonnegative integer power. Unlike the generalized series, this expansion terminates after finitely many terms.
Radius of convergence: The distance from a power series' center to the nearest boundary beyond which it diverges. For noninteger exponents, the binomial series converges for \(|x|<1\), with endpoints requiring separate tests.
Generalized binomial theorem: The extension of the binomial expansion to real or complex exponents, generally as an infinite series. This theorem states the expansion and its convergence conditions for generalized exponents.
Small-angle approximation: An approximation that simplifies trigonometric expressions when their angles are near zero. Binomial expansions approximate expressions such as \((1-x)^{1/2}\) that arise in geometric derivations.
Laurent series: A series in integer powers that may include negative exponents and represent functions near isolated singularities. The binomial series uses only nonnegative powers of \(x\), unlike expansions around punctured neighborhoods.
Ratio test: A convergence test that examines the limit of the ratio of successive term magnitudes. Applied to successive binomial terms, it establishes convergence inside the unit interval in absolute value.
Power series: An infinite sum of constant multiples of nonnegative integer powers of a variable. The binomial series is a particular power series centered at zero.

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