Coefficients and finite expansions

A positive integer power produces a finite polynomial. Choosing the second term from exactly \(r\) of the \(n\) factors explains the coefficient \(\binom nr\).

\((a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r,\qquad n\in\mathbb Z_{\ge0}\).

\(\binom nr=\dfrac{n!}{r!(n-r)!},\quad 0\le r\le n;\qquad 0!=1\).

The coefficients in successive rows of Pascal’s triangle are \(1\); \(1,1\); \(1,2,1\); \(1,3,3,1\); and \(1,4,6,4,1\). Each interior entry is the sum of the two entries above it.

\(\binom nr=\binom n{n-r}\). There are \(n+1\) terms before any like terms are collected.

For \((a-b)^n\), include the minus sign in \((-b)^r\). A finite expansion has no convergence restriction; any restrictions already present in the original expression still apply.