Calculus II · Unit 4B · lesson

The Binomial Series

Concept

Learning objectives

use the generalized binomial expansion for noninteger powers and determine its convergence interval.

The Binomial Series

Explanation

The finite binomial theorem has an infinite extension

For a nonnegative integer exponent, $(1+x)^m$ is a finite polynomial. For an arbitrary real exponent $\alpha$ , the same coefficient pattern continues indefinitely using generalized binomial coefficients. The resulting power series represents $(1+x)^\alpha$ for $|x|<1$ .

The expansion unifies many useful approximations, including square roots and reciprocal powers. Coefficients are best generated recursively rather than memorized. Endpoint behavior depends on $\alpha$ and requires separate analysis, so the standard statement emphasizes the open interval.

Bridge

The binomial theorem continues beyond integer powers

For a nonnegative integer exponent, the binomial expansion terminates. For a general real exponent $\alpha$ , the same coefficient pattern continues indefinitely and produces a power series for $(1+x)^\alpha$ when $|x|<1$ .

The generalized coefficients

\binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}

encode the changing numerator factors. When $\alpha$ is a nonnegative integer, one factor eventually becomes zero and the familiar finite theorem is recovered.

**Binomial partial sums approximate nonpolynomial powers**
The curve y equals square root of 1+x is compared with several binomial partial sums on an interval inside the radius of convergence.
Binomial partial sums approximate nonpolynomial powers. Partial sums approaching sqrt(1+x) or its reciprocal inside the radius.

Proof idea

The ratio test fixes the natural radius

For consecutive binomial terms, the magnitude ratio approaches $|x|$ . Thus the series converges absolutely for $|x|<1$ and diverges for $|x|>1$ ; endpoints require separate analysis depending on $\alpha$ .

Concept

Generalized binomial series

For $|x|<1$ ,

(1+x)^\alpha=\sum_{n=0}^{\infty}\binom{\alpha}{n}x^n, \quad \binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}.

Guided walkthrough

Square-root approximation

With $\alpha=1/2$ ,

\sqrt{1+x}=1+\frac12x-\frac18x^2+\frac1{16}x^3-\cdots.

At $x=0.04$ , the quadratic approximation gives $1.0198$ , close to $\sqrt{1.04}$ .

Worked example

Expand a reciprocal square root

For $\alpha=-1/2$ ,

(1+x)^{-1/2}=1-\frac12x+\frac38x^2-\frac5{16}x^3+\cdots, \qquad |x|<1.

The coefficients come from

\binom{-1/2}{2}=\frac{(-1/2)(-3/2)}{2!}=\frac38,

and

\binom{-1/2}{3}=\frac{(-1/2)(-3/2)(-5/2)}{3!}=-\frac5{16}.

Common mistake

The generalized coefficient is a falling product

The numerator uses $\alpha,\alpha-1,\alpha-2,\ldots$ , not repeated powers of $\alpha$ .

Interactive checku4b-binomial_series-01

Write the quadratic binomial approximation for $\sqrt{1+x}$ .

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Show hint

Use $\alpha=1/2$ .

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Exercise

Find the first four terms of $(1+x)^{-1/2}$ .

Exercise

Approximate $\sqrt{0.99}$ .

Exercise

Show that $\alpha=-1$ recovers the geometric series for $1/(1+x)$ .

Exercise

Discuss endpoint testing for one chosen value of $\alpha$ .

After the explanation

Use the section idea

Reading lens

Build new expansions from a small verified library using explicit substitutions, products, derivatives, or integrals.

Mental model

A standard series is a reusable identity with a domain, not a formula fragment detached from convergence.

Decision

Name the source series, apply one transformation at a time, and transform its interval alongside it.

Common trap

Recognizing a pattern but omitting the substitution's effect on the interval.

Check yourself

Can you reconstruct the expansion and its domain without memorizing the final line?

Source & rights

Original instruction with traceable references.

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