Pascal's Triangle

From wikiluntti

Introduction

Binomial expansion

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} (a+b)^0 &= 1 \\ (a+b)^1 &= a + b \\ (a+b)^2 &= a^2 + 2ab + b^2 \\ (a+b)^3 &= (a+b)(a+b)^2=(a+b)(a^2 + 2ab + b^2) \\ &= a^3 + 2a^2 b + ab^2 +ba^2 + 2ab^2+b^3 \\ &= a^3 + 3a^2 b + 3ab^2+b^3 \\ (a+b)^4 &= \cdots \end{align} }

Pascal's triangle

Pascal's Triangle

The coefficients of binomial expansion can be easily seen from the Pascal triangle. The number is a sum of the two numbers above it.

Pascal's triangle: Negative 1

This can be extended to negative numbers easily.

Pascal triangle extended to negative values

Now, instead of expanding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+b)^n} , we will use , where is a negative integer. The exponent of each terms grows when going to left. We get according to the Pascal triangle

And by Taylor series (expansion at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-1} Laurent series) we get