Binomial Theorem

From Conservapedia

The binomial theorem, also known as the binomial expansion, describes a method for expanding brackets containing two terms raised to a power as a polynomial.^[1] More specifically, it describes how to express a bracket of the form (x+a)ⁿ as a polynomial with terms b_kx^ka^n-k, for various coefficients b_k. When n is natural number, the binomial theorem produces a finite length polynomial, but can also be extended to non-integer n. For n not being a natural number, the binomial expansion of (x+a)ⁿ is:^[1]

\text{[math]}

Here, $\text{[math]}$ is the binomial coefficient and pronounced "n choose k". It is derived from combinatorics and equal to:

\text{[math]}

These coefficients for constant n form the rows of Pascal's triangle. The name comes from the Latin, bi-nomin, meaning two names (terms).^[2]

1 Proof
2 Generalisation to Non-Integers
3 Example
4 References
5 See also

Proof[edit]

The binomial theorem for integer n can easily be proved using proof by induction.^[3] Supposing the theorem is true for n=N. Then:

\text{[math]}

Multiplying out the bracket, and substituting j=k+1 gives:

\text{[math]}

Separating the k=0 and j=N+1 terms, the sums can be combined as:

\text{[math]}

Using a combinations identity, that $\text{[math]}$ ,^[3] this can be rewritten as:

\text{[math]}

Hence if the theorem holds for n=N, it must hold for n=N+1. As it is true for n=0, it is true for all natural numbers.

Generalisation to Non-Integers[edit]

When the binomial theorem is extended to non-integer n or n<0, the series it creates is infinite:^[1]

\text{[math]}

As it is an infinite series, it is often referred to as the "binomial series". The binomial coefficients are generalised so that n need not be a natural number as:^[4]

\text{[math]}

for k>0 and 1 for k=0. As this series contains an infinite number of terms, one must worry whether it converges or not. In the case of the binomial expansion, it always converges if |x/a|<1 or if n a natural number.^[1]

Other generalisations exist such as the multinomial expansion for brackets with more than two terms and the multi-binomial expansion where several binomial brackets are multiplied together.

Example[edit]

As an example, consider expanding (x+y)⁴. It would take a while doing it by hand. Using the binomial theorem, we quickly derive:

\text{[math]}

These coefficients form the 5th row of Pascal's triangle. As an example with negative n, consider (x+1)^-1:

\text{[math]}

which will converge for |x|<1.

References[edit]

↑ ^1.0 ^1.1 ^1.2 ^1.3 Binomial theorem from mathworld.wolfram.com
↑ Etymology from Merriam-Webster.com
↑ ^3.0 ^3.1 K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 3^rd ed., 2006
↑ Binomial series from tutorial.math.lamar.edu

Binomial Theorem

Contents

Proof[edit]

Generalisation to Non-Integers[edit]

Example[edit]

References[edit]

See also[edit]