Section 6.3 Taylor and Maclaurin Series
References.
- Calculus, Early Transcendentals by Stewart, Chapter 11, Section 10.
The last main ingredient in using power series is a more systematic way to find such a series for a particular function, like \(\cos x\) or \(\int e^{-x^2} dx\text{.}\)
We will follow a familiar strategy of
- devising a basic method,
- using this to get power series for the main elementary functions, and then
- learning how to combine these to get series for many other functions more easily, to avoid starting from scratch each time.
Consider a function \(f(x)\text{,}\) and let us try to find the coefficients \(c_0\text{,}\) \(c_1\) etc. giving a power series form
\begin{equation}
f(x) = c_0 + c_1x + c_2 x^2 + \cdots,\tag{6.3.1}
\end{equation}
at least for small enough \(|x|\text{.}\)
Evaluating at \(x=0\) gives
\begin{equation*}
c_0 = f(0)
\end{equation*}
To continue, differentiate, getting
\begin{equation*}
f'(x) = c_1 + 2 c_2 x + 3c_3 x^2 + \cdots
\end{equation*}
so again evaluating at \(x=0\) gives
\begin{equation*}
c_1 = f'(0)
\end{equation*}
The MacLaurin Series.
The pattern can continue of alternately differentiating and then evaluating at \(x=0\text{:}\) we get
\begin{equation}
f^{(n)}(x) = c_n n! + c_{n+1}(n+1)(n) \cdots (2) x + c_{n+2}(n+2)(n+1) \cdots (3) x^2 \cdots\tag{6.3.2}
\end{equation}
so that \(f^{(n)}(0) = c_n n!\) and
\begin{equation}
c_n = \frac{f^{(n)}(0)}{n!}\tag{6.3.3}
\end{equation}
Thus, it seems that \(f(x)\) is given by the power series
\begin{equation}
f(0) + f'(0) x + \frac{f''(0)}{2} x^2 \cdots + \frac{f^{(n)}(0)}{n!} x^n \cdots\tag{6.3.4}
\end{equation}
and this series is called the Maclaurin Series for function \(f\).
Find the Maclaurin series for \(f(x) = e^x\text{,}\) and its radius of convergence.
The Taylor Series.
Changing to the general power series with center \(a\) and using derivatives at \(x=a\) instead gives the Taylor Series for function \(f\) at center \(a\):
\begin{equation}
\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n
= f(a) + f'(a) (x-a)
\cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n \cdots\tag{6.3.5}
\end{equation}
But we must answer two questions:
- For which \(x\) values does such a series converge?
- When is the value that it converges to equal to \(f(x)\text{,}\) as we would hope?
Taylor Polynomials and Their Remainders.
To determine whether a Taylor series converges, and whether it gives the expected value, we look at partial sums and the remainders for them, which are now functions of \(x\text{.}\)
The \(N\)-th partial sum for a Taylor series is the Taylor Polynomial
\begin{equation}
T_N(x) = f(a) + f'(a) (x-a) \cdots + \frac{f^{(N)}(a)}{N!} (x-a)^N.\tag{6.3.6}
\end{equation}
The Taylor Series has a radius of convergence \(R\) (possibly 0 or \(\infty\text{!}\)); the error we care about is the difference between the \(N\)-th Taylor polynomial and the function \(f(x)\text{.}\)
This is called the remainder of the Taylor series; the function
\begin{equation}
R_N(x) = f(x) - T_N(x).\tag{6.3.7}
\end{equation}
Is the Remainder Small Enough? Taylor's Theorem and Taylor's Inequality.
We hope that as \(N \to \infty\text{,}\) \(T_N(x) \to f(x)\text{,}\) which is the same as the condition \(R_N(x) \to 0\text{.}\)
The main tool for checking this is to look at the size of the terms in the series through the size of the derivative values \(f^{(n)}(a)\text{:}\) it first helps to know that the remainder \(R_N(x)\) looks almost like the first term not used in the partial sum \(T_N(x)\text{:}\)
Theorem 6.3.2. Taylor's Theorem.
For a Taylor polynomial \(T_n(x)\) for \(f\) with center \(a\text{,}\) and any number \(x\) in the domain of \(f\text{,}\) there is a number \(c\) between \(x\) and \(a\) such that
\begin{equation}
R_N(x) = \frac{f^{(N+1)}(c)}{(N+1)!} (x-a)^{N+1}\tag{6.3.8}
\end{equation}
Then if the \(N\)-th derivatives do not get too big as \(N\) increases, we are done:
Theorem 6.3.3. Taylor's Inequality, and convergence to \(f(x)\text{,}\) sometimes.
If for some numbers \(M\) and \(d>0\text{,}\) \(|x-a| \leq d\) ensures that \(|f^{(n)}(x)| \leq M\text{,}\) for all \(n\text{,}\) then the remainder of the Taylor series for \(f\) with center \(a\) satisfies
\begin{equation}
|R_N(x)| \leq \frac{M}{(N+1)!} |x-a|^{N+1} \quad \text{ for } |x-a| \leq d.\tag{6.3.9}
\end{equation}
If this is true, then for \(|x-a| \leq d\text{,}\) \(|R_N(x)| \to 0\) as \(N \to \infty\text{,}\) so \(T_N(x) \to f(x)\text{,}\) so that the value of the Taylor Series is \(f(x)\text{:}\)
\begin{equation}
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n
= f(a) + f'(a) (x-a) + \frac{f''(a)}{2} (x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n \cdots\tag{6.3.10}
\end{equation}
A Warning: This theorem does not guarantee that the series has value \(f(x)\) for all \(x\) in its interval of convergence: this might be true only for some \(d\) less than \(R\text{.}\)
As an extreme case, the Taylor series for \(f(x)=e^{-1/x^2}\) (defining \(f(0)=0\) to make it continuous there) is \(0 + 0 \cdot x + 0 \cdot x^2 + \cdots = 0\text{.}\)
The good news is that, as we will soon see, the Taylor Series for any of our favorite elementary functions \(f(x)\) does converge to that function, for all \(x\) values in the interval of convergence of the series.
Checkpoint 6.3.4.
Prove that \(e^x\) is equal to its Maclaurin series for all \(x\text{.}\) That is: for any \(d>0\text{,}\) find a suitable \(M\) to use in Taylor's inequality.
Checkpoint 6.3.5.
Find the Taylor series for \(f(x) = \ln x\) with center \(a=1\text{.}\)
Then verify that the series converges to \(\ln x\) for all values \(x\) in its interval of convergence.
Building new Taylor Series from old.
As much as possible we wish to avoid working directly from the formula for the Taylor series, instead we seek to build new power series form ones already known:
Example 6.3.6.
Find the MacLaurin series for \(f(x) = \cos x\text{,}\) by differentiating the above MacLaurin series for \(\sin x\text{.}\)
Checkpoint 6.3.7.
Find a power series for \(f(x)=\ds\frac{\sin x}{x}\) (with value \(f(0)=1\) for continuity), again using the above MacLaurin series for \(\sin x\text{.}\)
Checkpoint 6.3.8.
Find a power series for \(\sinh x\text{,}\) using the above MacLaurin series for \(e^x\text{.}\)
Checkpoint 6.3.9.
Find the first three nonzero terms of the MacLaurin series for \(e^x \sin x\text{.}\)
Square Roots.
Having dealt with the most important transcendental functions, we need to deal with algebraic functions, and in particular roots: we start with the square root.
Checkpoint 6.3.10. A power series for the square root.
- Find the Taylor series for \(\sqrt{x}\) with center \(a=1\text{,}\)
- verify that its radius of convergence is \(R=1\text{,}\)
- and verify that for \(|x-1| < 1\text{,}\) the series converges to \(\sqrt{x}\text{.}\)
Study Guide.
Study Calculus Volume 2, Section 6.3 2 ; in particular
- The definitions of Taylor series and Maclaurin series and Taylor polynomials
- Theorem 6 on the uniqueness of Taylor series
- Theorem 7 on Taylor's Theorem with remainder
- Theorem 8 on the Convergence of Taylor series
- Examples 11, 12, 13, 14, 15, 16
- Checkpoints 10, 11, 12, 13, 14, 15
- and one or several exercises from each of the following groups: 116–123, 130 and 131, 132–135.
openstax.org/books/calculus-volume-2/pages/6-3openstax.org/books/calculus-volume-2/pages/6-3
