Algebra and Substitution Algebra and Substitution

Introduction

We have used differentiation and integration to derive new Taylor series.

We will now use algebraic manipulations and substitutions.

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
.