Binomial Series Expansion

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Understanding Binomial Expansion

A binomial expression is an algebraic expression containing two terms. For example, $a+b$, $2x-y$, or $x^2+3$.

The binomial theorem provides a formula for expanding expressions of the form $(a+x)^n$, where $n$ is a positive integer. This process can be visualized and aided by Pascal's Triangle for smaller values of $n$. For more general cases, including negative or fractional $n$, we use the Binomial Series.

Expand $(a+b)^n$ using Pascal's Triangle

Pascal's Triangle provides the coefficients for the expansion of $(a+b)^n$ when $n$ is a non-negative integer. Each row $n$ (starting from row 0) gives the coefficients for the power $n$.

$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

Expand $(a+x)^n$ using Binomial Series (any $n$)

For any real number $n$, the expansion of $(1+y)^n$ is given by the Binomial Series: $(1+y)^n = 1 + ny + \frac{n(n-1)}{2!}y^2 + \frac{n(n-1)(n-2)}{3!}y^3 + \dots$

This series is infinite if $n$ is not a positive integer and converges if $|y| < 1$. To expand $(a+x)^n$, we use $a^n(1 + \frac{x}{a})^n$. Valid if $|\frac{x}{a}| < 1$.

Find the $r^{th}$ Term of $(a+x)^n$

The general term ($(k+1)^{th}$ term) is $T_{k+1} = \binom{n}{k} a^{n-k} x^k$. So, the $r^{th}$ term is $T_r = \binom{n}{r-1} a^{n-(r-1)} x^{r-1}$.

If $n$ is not a positive integer, $\binom{n}{k} = \frac{n(n-1)\dots(n-k+1)}{k!}$ for $k > 0$, $\binom{n}{0}=1$.

General Binomial Expansion Formulas

Expansion of $(a+x)^n$ for positive integer $n$:

The Binomial Theorem states:

$(a+x)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} x^k$

$= \binom{n}{0}a^n x^0 + \binom{n}{1}a^{n-1}x^1 + \binom{n}{2}a^{n-2}x^2 + \dots + \binom{n}{n}a^0 x^n$

$= a^n + n a^{n-1}x + \frac{n(n-1)}{2!}a^{n-2}x^2 + \dots + nax^{n-1} + x^n$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

Expansion of $(1+x)^n$ for positive integer $n$:

A special case of the above, where $a=1$:

$(1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k$

$= \binom{n}{0}x^0 + \binom{n}{1}x^1 + \binom{n}{2}x^2 + \dots + \binom{n}{n}x^n$

$= 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots + x^n$