# Binomial Series

Real numbers: $$x$$, $$n$$
Whole numbers: $$m$$, $$n$$
Number of $$m$$-combinations of $$n$$ elements: $$\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right)$$
1. The binomial series is the Maclaurin series expansion of the function $${\left( {1 + x} \right)^n}$$ and, in general, is written as
$${\left( {1 + x} \right)^n} =$$ $$\sum\limits_{m = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m}} =$$ $$1 + \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right)x$$ $$+\; \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^2} + \left( {\begin{array}{*{20}{c}} n\\ 3 \end{array}} \right){x^3} + \ldots$$ $$+\;\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m} + \ldots ,$$
where $$\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right)$$ are the binomial coefficients, $$m$$ is a whole number, $$x$$ is a real (or complex) variable, $$n$$ is a real (or complex) power.
2. The binomial coefficients are expressed by the formula
$$\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right) =$$ $${\large\frac{{n\left( {n – 1} \right) \ldots \left( {n – m + 1} \right)}}{{m!}}\normalsize}$$, where $$0 \le m \le n$$.
3. The binomial series converges under the following conditions (assuming that $$x$$ and $$n$$ are real numbers):
$$-1 \lt x \lt 1$$, if $$n \lt -1$$;
$$-1 \lt x \le 1$$, if $$-1 \lt n \lt 0$$;
$$-1 \le x \le 1$$, if $$n \gt 0$$.
4. Binomial theorem
In the case of nonnegative integer powers $$n$$, the binomial series is a finite sum of $${n + 1}$$ terms and is referred to as binomial theorem:
$${\left( {1 + x} \right)^n} =$$ $$\sum\limits_{m = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m}} =$$ $$1 + \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right)x + \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^2}$$ $$+\;\left( {\begin{array}{*{20}{c}} n\\ 3 \end{array}} \right){x^3} + \ldots$$ $$+\; \left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right){x^m} + \ldots$$ $$= {1 + nx + {\large\frac{{n\left( {n – 1} \right)}}{{2!}}\normalsize} {x^2} + \ldots }$$ $$+\;{ {x^n}}$$
5. Binomial coefficients as the number of combinations The coefficients in the binomial theorem are equal to the number of $$m$$-combinations of $$n$$ elements:
$$\left( {\begin{array}{*{20}{c}} n\\ m \end{array}} \right) =$$ $${\large\frac{{n!}}{{m!\left( {n – m} \right)!}}\normalsize} =$$ $${\large\frac{{n\left( {n – 1} \right)\left( {n – 1} \right) \ldots \left( {n – m + 1} \right)}}{{m!}}\normalsize}.$$
6. Some common binomial expansions:
7. $${\large\frac{1}{{1 + x}}\normalsize} =$$ $$1 – x + {x^2}$$ $$-\; {x^3} + \ldots,$$ $$\left| x \right| \lt 1$$
8. $${\large\frac{1}{{1 – x}}\normalsize} =$$ $$1 + x + {x^2}$$ $$+\; {x^3} + \ldots,$$ $$\left| x \right| \lt 1$$
9. $$\sqrt {1 + x} =$$ $$1 + {\large\frac{x}{2}\normalsize} – {\large\frac{{{x^2}}}{{2 \cdot 4}}\normalsize}$$ $$+\;{\large\frac{{1 \cdot 3{x^3}}}{{2 \cdot 4 \cdot 6}}\normalsize}$$ $$-\;{\large\frac{{1 \cdot 3 \cdot 5{x^4}}}{{2 \cdot 4 \cdot 6 \cdot 8}}\normalsize} + \ldots,$$ $$\left| x \right| \le 1$$
10. $$\sqrt[\large 3\normalsize]{{1 + x}} =$$ $$1 + {\large\frac{x}{3}\normalsize} – {\large\frac{{1 \cdot 2{x^2}}}{{3 \cdot 6}}\normalsize}$$ $$+\;{\large\frac{{1 \cdot 2 \cdot 5{x^3}}}{{3 \cdot 6 \cdot 9}}\normalsize}$$ $$-\;{\large\frac{{1 \cdot 2 \cdot 5 \cdot 8{x^4}}}{{3 \cdot 6 \cdot 9 \cdot 12}}\normalsize} + \ldots,$$ $$\left| x \right| \le 1$$