Binomial Series

The central idea

For any real or complex exponent $\alpha$, $(1+x)^\alpha=\sum_{n=0}^{\infty}{\alpha\choose n}x^n$ for $|x|<1$, where ${\alpha\choose0}=1$ and ${\alpha\choose n}=\alpha(\alpha-1)\cdots(\alpha-n+1)/n!$. For nonnegative integer $\alpha$, the series terminates and is valid for all $x$.

Definitions, hypotheses, and notation

A convenient coefficient recurrence is $c_{n+1}=c_n(\alpha-n)/(n+1)$ with $c_0=1$. It reduces arithmetic and reveals when the expansion terminates: for a nonnegative integer exponent, the numerator eventually becomes zero. For other exponents, the ratio test gives radius one.

Endpoint behavior depends on $\alpha$ and cannot be read from the radius alone. Scaling is often needed: for example, $\sqrt{9+h}=3(1+h/9)^{1/2}$, so the series variable is $h/9$ and convergence requires $|h|<9$, not $|h|<1$.

Conceptual meaning

The familiar finite binomial theorem extends to noninteger powers through an infinite power series. Coefficients are generated by repeated differentiation at zero. Convergence is guaranteed inside radius one; endpoints require separate analysis.

A dependable method and decision rule

1. Rewrite the expression in the form $(1+x)^\alpha$.
2. Compute generalized binomial coefficients recursively or from the product formula.
3. Attach the correct power $x^n$ and signs.
4. State the condition $|x|<1$ before using the infinite expansion.
5. For approximation, retain enough terms and estimate the first neglected contribution.

Fully worked example

Graphical or geometric meaning

Partial-sum polynomials match increasingly many derivatives of $(1+x)^\alpha$ at zero. Near zero their graphs are nearly indistinguishable; near the convergence boundary more terms are generally needed.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate or square a short approximation when feasible.
• Verify the first coefficient is $1$ and the linear coefficient is $\alpha$.
• Check that the evaluation point lies within the convergence interval.

The exponent controls the coefficient pattern

For real $\alpha$, $(1+x)^\alpha=\sum_{n=0}^\infty\binom{\alpha}{n}x^n$ for $|x|<1$, where $\binom{\alpha}{n}=\alpha(\alpha-1)\cdots(\alpha-n+1)/n!$. When $\alpha$ is a nonnegative integer, the coefficients eventually vanish and the finite binomial theorem results. Otherwise the expansion is generally infinite, and $x=\pm1$ requires separate testing. To expand $(a+bx)^\alpha$, factor out $a$ and rewrite the remainder as $1+u$; the condition becomes $|u|<1$. Check the constant and linear coefficients from the function value and derivative at the center. This catches a missing scale factor before later coefficients compound it.

Practice

1. Give the first three terms of $(1+x)^{-1}$.
2. Give the coefficient of $x^2$ in $(1+x)^\alpha$.
3. What happens for $\alpha=3$?

Connections and next steps

• Maclaurin Series
• Power Series
• Interval of Convergence