9.9 Taylor Series and Maclaurin Series
Taylor Series and Maclaurin Series
If a function $f$ has derivatives of all orders at $x=c$, then the series
$
\begin{aligned}
& \sum_{n=0}^{\infty} \frac{f^{(n)}(c)}{n !}(x-c)^n \\
& =f(c)+f^{\prime}(c)(x-c)+\frac{f^{\prime \prime}(c)}{2 !}(x-c)^2+\frac{f^{\prime \prime \prime}(c)}{3 !}(x-c)^3+\cdots+\frac{f^{(n)}(c)}{n !}(x-c)^n+\cdots
\end{aligned}
$
is called the Taylor series for $\boldsymbol{f}(\boldsymbol{x})$ at $c$. Moreover, if $c=0$, then the series is called the Maclaurin series for $\boldsymbol{f}$.
Example 1
- Find the Maclaurin series for the function $f(x)=\ln (1+x)$.
▶️Answer/Explanation
Solution
$\begin{array}{ll}f(x)=\ln (1+x) & f(0)=0 \\ f^{\prime}(x)=\frac{1}{1+x} & f^{\prime}(0)=1 \\ f^{\prime \prime}(x)=-\frac{1}{(1+x)^2} & f^{\prime \prime}(0)=-1 \\ f^{\prime \prime \prime}(x)=\frac{2}{(1+x)^3} & f^{\prime \prime \prime}(0)=2 \\ f^{(4)}(x)=-\frac{6}{(1+x)^4} & f^{(4)}(0)=-6\end{array}$
$\begin{aligned} & \ln (1+x)=f(0)+f^{\prime}(0) x+\frac{f^{\prime \prime}(0)}{2 !} x^2+\frac{f^{\prime \prime \prime}(0)}{3 !} x^3+\cdots+\frac{f^{(n)}(0)}{n !} x^n+\cdots \\ & =0+1 \cdot x+\frac{-1}{2 !} x^2+\frac{2}{3 !} x^3+\frac{-6}{4 !} x^4+\cdots+\frac{(-1)^{n-1}(n-1) !}{n !} x^n+\cdots \cdot \\ & =x-\frac{1}{2} x^2+\frac{1}{3} x^3-\frac{1}{4} x^4+\cdots \cdot+\frac{(-1)^{n-1}}{n} x^n+\cdots \cdot\end{aligned}$
Example 2
- Let $f$ be a function having derivatives of all orders. The fourth degree Taylor polynomial for $f$ about $x=1$ is given
▶️Answer/Explanation
solution
$
T(x)=4+3(x-1)-6(x-1)^2+7(x-1)^3-4(x-1)^4 \text {. }
$
Find $f(1), f^{\prime}(1), f^{\prime \prime}(1), f^{\prime \prime \prime}(1)$ and $f^{(4)}(1)$.
$\begin{array}{ll}f(1)=T(1)=4 & f^{\prime}(1)=3 \\ \frac{f^{\prime \prime}(1)}{2 !}=-6 \Rightarrow f^{\prime \prime}(1)=-12 & \frac{f^{\prime \prime \prime}(1)}{3 !}=7 \Rightarrow f^{\prime \prime \prime}(1)=42 \\ \frac{f^{(4)}(1)}{4 !}=-4 \Rightarrow f^{(4)}(1)=-96 & \end{array}$
Direct computation of the Taylor or Maclaurin coefficients is usually a tedious procedure. The easiest way to find a Taylor or Maclaurin series is to develop a power series from a list of elementary functions. From the list of power series for elementary functions, you can develop power series for other functions by the operations of addition, subtraction, multiplication, division, differentiation, integration, or composition with known power series.
Power Series for Elementary Functions
Function Interval of Convergence
$
\begin{aligned}
& \frac{1}{x}=1-(x-1)+(x-1)^2-(x-1)^3+\cdots \cdot+(-1)^n(x-1)^n+\cdots \cdot 0<x<2 \\
& \frac{1}{1-x}=1+x+x^2+x^3+\cdots+x^n+\cdots \cdot \quad-1<x<1 \\
& \ln x=(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\cdots+\frac{(-1)^{n-1}(x-1)^n}{n}+\cdots \cdot 0<x \leq 2 \\
& e^x=1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots+\frac{x^n}{n !}+\cdots \cdot \quad-\infty<x<\infty \\
& \sin x=x-\frac{x^3}{3 !}+\frac{x^5}{5 !}-\frac{x^7}{7 !}+\cdots+\frac{(-1)^n x^{2 n+1}}{(2 n+1) !}+\cdots \cdot \quad-\infty<x<\infty \\
& \cos x=1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\cdots+\frac{(-1)^n x^{2 n}}{(2 n) !}+\cdots \cdot \quad-\infty<x<\infty \\
& \tan ^{-1} x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots+\frac{(-1)^n x^{2 n+1}}{2 n+1}+\cdots \cdot \quad-1 \leq x \leq 1 \\
&
\end{aligned}
$
Multiplication of Power Series
Power series can be multiplied the way we multiply polynomials. We usually find only the first few terms because the calculations for the later terms become tedious and the initial terms are the most important ones.
Example 4
- Find the Maclaurin series for the function $f(x)=\cos x^2$.
▶️Answer/Explanation
Solution
$\begin{aligned} g(x) & =\cos x=1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\cdots+\frac{(-1)^n x^{2 n}}{(2 n) !}+\cdots \\ f(x) & =\cos x^2 \\ & =g\left(x^2\right) \\ & =1-\frac{\left(x^2\right)^2}{2 !}+\frac{\left(x^2\right)^4}{4 !}-\frac{\left(x^2\right)^6}{6 !}+\cdots+\frac{(-1)^n\left(x^2\right)^{2 n}}{(2 n) !}+\cdots \\ & =1-\frac{x^4}{2 !}+\frac{x^8}{4 !}-\frac{x^{12}}{6 !}+\cdots+\frac{(-1)^n x^{4 n}}{(2 n) !}+\cdots\end{aligned}$
Example 5
- Find the Maclaurin series for the function $f(x)=x^2 e^x-x^2$
▶️Answer/Explanation
Solution
Use the series $e^x$.
$
e^x=1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots+\frac{x^n}{n !}+.
$
Multiply $e^x$ by $x^2$.
$
\begin{aligned}
x^2 e^x & =x^2\left(1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots+\frac{x^n}{n !}+\cdots\right) \\
& =x^2+x^3+\frac{x^4}{2 !}+\frac{x^5}{3 !}+\cdots \cdot+\frac{x^{n+2}}{n !}+\cdots \cdot
\end{aligned}
$
Subtract $x^2$ from each side.
$
\begin{aligned}
& x^2 e^x-x^2 \\
& =\left(x^2+x^3+\frac{x^4}{2 !}+\frac{x^5}{3 !}+\cdots+\frac{x^{n+2}}{n !}+\cdots\right)-x^2 \\
& =x^3+\frac{x^4}{2 !}+\frac{x^5}{3 !}+\cdots+\frac{x^{n+2}}{n !}+\cdots \cdot
\end{aligned}
$
Example 6
- Find the first three nonzero terms in the Maclaurin series for $e^x \cos x$.
▶️Answer/Explanation
Solution
Use the power series for $e^x$ and $\cos x$ in the table.
Use the power series for $e^x$ and $\cos x$ in the table.
$
\begin{aligned}
e^x \cos x= & \left(1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots\right)\left(1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\cdots\right) \\
= & \left(1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots\right)(1)+\left(1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots\right)\left(-\frac{x^2}{2 !}\right)+\cdots \\
= & 1+x+\frac{x^2}{2 !}+\frac{x^3}{3 !}+\cdots \\
& \quad-\frac{x^2}{2 !}-\frac{x^3}{2 !}-\cdots \\
= & 1+x-\frac{x^3}{3}+\cdots
\end{aligned}
$
Example7
- A series expansion of $\frac{\arctan x}{x}$ is
(A) $1-\frac{x}{3}+\frac{x^3}{5}-\frac{x^5}{7}+\cdots$
(B) $1-\frac{x^2}{3}+\frac{x^4}{5}-\frac{x^6}{7}+\cdots$
(C) $1-\frac{x^2}{2 !}+\frac{x^4}{4 !}-\frac{x^6}{6 !}+\cdots$
(D) $x-\frac{x^3}{3}+\frac{x^4}{5}-\frac{x^6}{7}+\cdots$
▶️Answer/Explanation
Ans:B
Example 8
- The coefficient of $x^3$ in the Taylor series for $e^{-2 x}$ about $x=0$ is
(A) $-\frac{4}{3}$
(B) $-\frac{2}{3}$
(C) $-\frac{1}{3}$
(D) $\frac{4}{3}$
▶️Answer/Explanation
Ans:A
Example9
- A function $f$ has a Maclaurin series given by $-\frac{x^4}{3 !}+\frac{x^6}{5 !}-\frac{x^8}{7 !}+\cdots+\frac{(-1)^n x^{2 n+2}}{(2 n+1) !}+\cdots$. Which of the following is an expression for $f(x)$ ?
(A) $x^3 e^x-x^2$
(B) $x \ln x-x^2$
(C) $\tan ^{-1} x-x$
(D) $x \sin x-x^2$
▶️Answer/Explanation
Ans:D