Algebraic Properties
Addition and Subtraction
f(x) = \sum_{n=0}^{\infty} a_n x^n \quad g(x) = \sum_{n=0}^{\infty} b_n x^n \\
\begin{align*}\\
f(x) \pm g(x) &= \sum_{n=0}^{\infty} a_n x^n + \sum_{n=0}^{\infty} b_n x^n \\\\
&= \left[ a_0 + a_1 x + a_2 x^2 + \cdots \right] \pm \left[ b_0 + b_1 x + b_2 x^2 + \cdots \right] \\\\
&= (a_0 \pm b_0) + (a_1 \pm b_1) x + (a_2 \pm b_2) x^2 + \cdots \\\\
&= \sum_{n=0}^{\infty} (a_n \pm b_n) x^n
\end{align*}
Multiplication
f(x) = \sum_{n=0}^{\infty} a_n x^n \quad g(x) = \sum_{n=0}^{\infty} b_n x^n \\
\begin{align*}\\
f(x) g(x) &= \left( \sum_{n=0}^{\infty} a_n x^n \right) \left( \sum_{m=0}^{\infty} b_m x^m \right) \\\\
&= \left[ a_0 + a_1 x + a_2 x^2 + \cdots \right] \left[ b_0 + b_1 x + b_2 x^2 + \cdots \right] \\\\
&= a_0 b_0 + (a_0 b_1 + a_1 b_0) x + (a_0 b_2 + a_1 b_1 + a_2 b_0) x^2 + \cdots \\\\
&= \sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} a_k b_{n-k} \right) x^n
\end{align*}
Substitution
\begin{align*}
f(x) = \sum_{n=0}^{\infty} a_n x^n \\\\
f(g(x)) = \sum_{n=0}^{\infty} a_n g(x)^n
\end{align*}
Examples
Example 1:
Find the Maclaurin series for
f(x) = \cos(2x)
by substitution into the Maclaurin series for
\cos(x)
.
\begin{align*}
\cos(x) &= \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \\
\cos(2x) &= \sum_{n=0}^{\infty} \frac{(-1)^n (2x)^{2n}}{(2n)!} \\
&= \sum_{n=0}^{\infty} \frac{(-1)^n 2^{2n} x^{2n}}{(2n)!} \\
&= \sum_{n=0}^{\infty} \frac{(-1)^n 4^{n} x^{2n}}{(2n)!} \\
&= \sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!}
\end{align*}
The Maclaurin series for
\cos(2x)
is
\sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!}
.
Example 2:
Find the Maclaurin series for
f(x) = (e^x)^2
by substitution into the Maclaurin series for
e^x
.
\begin{align*}
e^x &= \sum_{n=0}^{\infty} \frac{x^n}{n!} \\
(e^x)^2 &= \sum_{n=0}^{\infty} \frac{(x^2)^n}{n!} \\
&= \sum_{n=0}^{\infty} \frac{x^{2n}}{n!}
\end{align*}
The Maclaurin series for
(e^x)^2
is
\sum_{n=0}^{\infty} \frac{x^{2n}}{n!}
.
Example 3:
Find the Maclaurin series for
f(x) = \sin^2(x) + \cos^2(x)
.
\begin{align*}
\cos(2x) &= \sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!} \\\\
\sin^2(x) &= \frac{1 - \cos(2x)}{2} \\
&= \frac{1}{2} - \frac{1}{2} \cos(2x) \\
&= \frac{1}{2} - \frac{1}{2} \sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!} \\\\
\cos^2(x) &= \frac{1 + \cos(2x)}{2} \\
&= \frac{1}{2} + \frac{1}{2} \cos(2x) \\
&= \frac{1}{2} + \frac{1}{2} \sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!} \\\\
\sin^2(x) + \cos^2(x) &= \frac{1}{2} - \frac{1}{2} \sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!} + \frac{1}{2} + \frac{1}{2} \sum_{n=0}^{\infty} \frac{(-4)^n x^{2n}}{(2n)!} \\
&= \frac{1}{2} + \frac{1}{2} \\
&= 1
\end{align*}
The Maclaurin series for
\sin^2(x) + \cos^2(x)
is
1
.