The binomial series

The Binomial Theorem provides us with a way to expand any power of a binomial without needing to multiply it out.
The general formula for the Binomial Theorem is: (a+b)^n = ∑ [(nCk) * a^(n-k) * b^k] where C stands for combination and ∑ stands for summation. These are worked out over all values of k from 0 to n.
(nCk) is the binomial coefficient and can be calculated using factorials: n! / [(n-k)!k!]. The exclamation mark represents a factorial, which means the product of an integer and all the integers below it. For example, 5! = 5x4x3x2x1.
The Binomial Theorem can also be used for negative or fractional powers if ** b/a < 1**.
The series would then become infinite as there will always be further terms. However, these additional terms become so small that they can be disregarded for practical purposes, hence (a+b)^n is approximated by the first few terms of the series.

For negative or fractional n, the Binomial Expansion becomes an infinite series: (1+x)^n = 1 + nx + n(n-1)/2! * x^2 + … and so on. It only converges if **

<1**.

Pascal’s Triangle

Pascal’s Triangle is a simple, yet valuable tool for calculating the coefficients in the binomial expansion. The row number represents the power of the binomial, and the numbers in the row give the binomial coefficients.
To generate this triangle, you start with the first line as [1], the second line as [1, 1] and you build additional rows where each number is the sum of the two numbers directly above it. For example, the third line will be [1, 2, 1].

Applications of the Binomial Theorem

In Finance: The binomial model in finance is a method of evaluation of options.
For Probability: The binomial distribution, based on the binomial theorem, is a concept in statistics used to predict the outcome of events.
In Computer Science: Algorithms based on the binomial coefficient are used for computer programming.