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Section A.2 Binomial and Power Series Expansion
-
Binomial Theorem:
\begin{equation*}
(a+b)^{n} = \left[ {}^{n}C_{0} a^n + {}^{n}C_{1} a^{n-1}b +{}^{n}C_{2} a^{n-2}b^2 +\cdots +{}^{n}C_{n} a^{n-n}b^n \right]
\end{equation*}
\begin{equation*}
or, \, (a+b)^{n} = a^{n}+na^{n-1}b+\frac{n(n-1)a^{n-2}b^{2}}{2!}+\frac{n(n-1)(n-2)a^{n-3}b^{3}}{3!}+\cdots
\end{equation*}
-
Power Series:
\begin{equation*}
(1+x)^{n} =1+nx+\frac{n(n-1)x^{2}}{2!}+\frac{n(n-1)(n-2)x^{3}}{3!}+\cdots \quad (|x| \lt 1)
\end{equation*}
\begin{equation*}
e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots \quad (\forall x)
\end{equation*}
\begin{equation*}
\ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots \quad (|x| \lt 1) or, (-1\lt x\leq 1.)
\end{equation*}
\begin{equation*}
\sin x= x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots \quad (\forall x)
\end{equation*}
\begin{equation*}
\cos x= 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots \quad (\forall x)
\end{equation*}
\begin{equation*}
\tan x= x+\frac{x^{3}}{3}+\frac{2x^{5}}{15}+\frac{17x^{7}}{315}+\cdots \quad (|x| \lt \frac{\pi}{2})
\end{equation*}
