Binomial Series for MATH 141

The Binomial Series

The binomial series extends the familiar Binomial Theorem to any exponent—not just positive integers. This lets us expand expressions like $\sqrt{1+x}$, $\frac{1}{(1+x)^3}$, or $(1+x)^{2/3}$ as infinite power series.

The Binomial Theorem (Review)

For positive integer $n$:

This is a finite sum with $n+1$ terms.

Example: $(1+x)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4$

The Binomial Series (General Case)

For any real exponent $k$ (including fractions, negatives, etc.):

Where the generalized binomial coefficient is:

$$\binom{k}{n} = \frac{k(k-1)(k-2)\cdots(k-n+1)}{n!}$$

Convergence: This series converges for $|x| < 1$.

Why Non-Integer Exponents Give Infinite Series

When $k$ is a positive integer, the coefficients eventually become zero:

• $\binom{4}{5} = \frac{4 \cdot 3 \cdot 2 \cdot 1 \cdot 0}{5!} = 0$

But when $k$ is NOT a positive integer, the numerator never hits zero:

• $\binom{1/2}{5} = \frac{(1/2)(-1/2)(-3/2)(-5/2)(-7/2)}{5!} \neq 0$

The terms keep going forever!

Common Binomial Series to Know

Square Root: $(1+x)^{1/2} = \sqrt{1+x}$

Reciprocal Square Root: $(1+x)^{-1/2} = \frac{1}{\sqrt{1+x}}$

Reciprocal: $(1+x)^{-1} = \frac{1}{1+x}$

$$\frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - \cdots$$

(This is just the geometric series!)

Cube Root: $(1+x)^{1/3} = \sqrt[3]{1+x}$

How to Compute Binomial Coefficients

For $\binom{k}{n}$, multiply $n$ terms in the numerator starting from $k$ and decreasing by 1:

$$\binom{k}{n} = \frac{k \cdot (k-1) \cdot (k-2) \cdots (k-n+1)}{n!}$$

Example: Find $\binom{1/2}{3}$

Example: Find $\binom{-2}{3}$

$$\binom{-2}{3} = \frac{(-2)(-3)(-4)}{3!} = \frac{-24}{6} = -4$$

Handling $(a + x)^k$ When $a \neq 1$

The binomial series formula requires the form $(1 + \text{something})^k$.

Strategy: Factor out $a$ first.

$$(a + x)^k = a^k\left(1 + \frac{x}{a}\right)^k$$

Then apply the binomial series to $\left(1 + \frac{x}{a}\right)^k$ and multiply by $a^k$.

Convergence changes: The series converges when $\left|\frac{x}{a}\right| < 1$, i.e., $|x| < |a|$.

Common Mistakes and Misunderstandings

❌ Mistake: Forgetting the pattern in the generalized coefficient

Wrong: $\binom{1/2}{3} = \frac{(1/2)(1/2)(1/2)}{3!}$

Why it's wrong: Each factor in the numerator must decrease by 1, not stay the same.

Correct: $\binom{1/2}{3} = \frac{(1/2)(-1/2)(-3/2)}{3!}$

❌ Mistake: Not converting to $(1 + \text{something})$ form

Wrong: Trying to expand $(3+x)^{1/2}$ directly using $1 + kx + \frac{k(k-1)}{2!}x^2 + \cdots$

Why it's wrong: The binomial series formula only works for $(1+u)^k$, not $(a+u)^k$.

Correct: Factor first: $(3+x)^{1/2} = 3^{1/2}(1 + \frac{x}{3})^{1/2} = \sqrt{3}(1 + \frac{x}{3})^{1/2}$, then expand $(1 + \frac{x}{3})^{1/2}$.

❌ Mistake: Expecting an infinite series for integer exponents

Wrong: Writing $(1+x)^3 = 1 + 3x + \frac{3(2)}{2!}x^2 + \frac{3(2)(1)}{3!}x^3 + \frac{3(2)(1)(0)}{4!}x^4 + \cdots$

Why it's wrong: When $k$ is a non-negative integer, the series terminates! The $x^4$ term has a factor of $(3-3) = 0$ in the numerator.

Correct: $(1+x)^3 = 1 + 3x + 3x^2 + x^3$ — a finite polynomial with 4 terms.

❌ Mistake: Using the series outside its radius of convergence

Wrong: Using $(1+x)^{-1} = 1 - x + x^2 - x^3 + \cdots$ to compute $\frac{1}{1+2} = 1 - 2 + 4 - 8 + \cdots$

Why it's wrong: The series only converges for $|x| < 1$. Here $x = 2$, so the series diverges.

Correct: Only use the binomial series when $|x| < 1$. For $x = 2$, just compute $\frac{1}{3}$ directly.