Pascal's Triangle

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 $(1+x)^{n}$, where $n$ is a negative integer. The exponent of each terms grows when going to left. We get according to the Pascal triangle

${\begin{aligned}(1+x)^{-1}&=1-x+x^{2}-x^{3}+\cdots \\(1+x)^{-2}&=1-2x+3x^{2}-4x^{3}+\cdots \\(1+x)^{-3}&=1-3x+6x^{2}-\cdots \end{aligned}}$

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

• https://www.wolframalpha.com/input?i=series%28+%281%2Bx%29%5E%28-1%29+%29
• https://www.wolframalpha.com/input?i=series%28+%281%2Bx%29%5E%28-2%29+%29
• https://www.wolframalpha.com/input?i=series%28+%281%2Bx%29%5E%28-3%29+%29