23. Taylor Series

a.1. Taylor Series Definition

In the previous chapter, we saw that a power series converges to a function on its interval of convergence. We now turn this around. Given a function, \(f(x)\) which is defined on an open interval containing \(x=a\), does there exist a power series centered at \(x=a\) which converges to \(f(x)\) on some interval? Such a series is called the Taylor series for \(f(x)\) about \(x=a\).

Given a function, \(f\), assume there is a power series centered at \(x=a\) which converges to that function. Find the power series.

To find a formula for the series, we write the series as: \[\begin{aligned} f(x)&=\sum_{n=0}^\infty c_n(x-a)^n \\ &=c_0+c_1(x-a)+c_2(x-a)^2+c_3(x-a)^3+c_4(x-a)^4+\cdots \qquad\text{(0)} \end{aligned}\] and try to find the coefficients, \(c_n\).

To find \(c_0\):  Set \(x=a\) in equation \(0\): \[ f(a)=c_0+c_1(a-a)+c_2(a-a)^2+c_3(a-a)^3+c_4(a-a)^4+\cdots=c_0 \] since all terms are zero except the first. So we conclude \[ c_0=f(a) \] To find \(c_1\):  Differentiate equation \((0)\) to obtain \[ f'(x)=c_1+2c_2(x-a)+3c_3(x-a)^2+4c_4(x-a)^3+\cdots \qquad\qquad\text{(1)} \] Set \(x=a\) in equation \((1)\): \[ f'(a)=c_1+2c_2(a-a)+3c_3(a-a)^2+4c_4(a-a)^3+\cdots=c_1 \] since all terms are zero except the first. So we conclude \[ c_1=f'(a) \] To find \(c_2\):  Differentiate equation \((1)\) to obtain \[ f''(x)=2c_2+3\cdot 2c_3(x-a)+4\cdot 3c_4(x-a)^2+\cdots \qquad\qquad\text{(2)} \] Set \(x=a\) in equation \((2)\): \[ f''(a)=2c_2+3\cdot 2c_3(a-a)+4\cdot 3c_4(a-a)^2+\cdots=2c_2 \] since all terms are zero except the first. So we conclude \[ c_2=\dfrac{1}{2}\,f''(a) \] To find \(c_3\):  Differentiate equation \((2)\) to obtain \[ f'''(x)=3\cdot 2c_3+4\cdot 3\cdot2c_4(x-a)+\cdots \qquad\qquad\text{(3)} \] Set \(x=a\) in equation \((3)\): \[ f'''(a)=3\cdot 2c_3+4\cdot 3\cdot2c_4(a-a)+\cdots =3!\,c_3 \] since all terms are zero except the first. So we conclude \[ c_3=\dfrac{1}{3!}\,f'''(a) \] To find \(c_4\), \(c_5\), etc.:  If you proceed in this manner, you find \[ c_4=\dfrac{1}{4!}\,f^{(4)}(a),\qquad c_5=\dfrac{1}{5!}\,f^{(5)}(a), \qquad \text{etc.} \] where \(f^{(n)}(a)\) denotes the \(n^\text{th}\) derivative of \(f(x)\) evaluated at \(x=a\). In general, we conclude \[ c_n=\dfrac{1}{n!}\,f^{(n)}(a) \] Note: \(f^{(0)}(x)\)is the \(0^\text{th}\) derivative of \(f(x)\) which is just \(f(x)\) itself.

By substituting these results into equation \((0)\), we conclude:

The Taylor series for a function \(f(x)\) at \(x=a\) is the series: \[\begin{aligned} Tf(x) &=\sum_{n=0}^\infty \dfrac{f^{(n)}(a)}{n!}(x-a)^n \\ &=f(a)+f'(a)(x-a)+\dfrac{f''(a)}{2}(x-a)^2 \\[2 pt] &\quad+\dfrac{f^{(3)}(a)}{3!}(x-a)^3+\cdots+\dfrac{f^{(n)}(a)}{n!}(x-a)^n+\cdots \end{aligned}\] This series is also called the Taylor series for \(f(x)\) near, about, around or centered at \(x=a\).

If the function \(f(x)\) is the limit of a power series centered at \(x=a\) then the series is its Taylor series: \[ f(x)=Tf(x) \]

If the Taylor series is centered at \(x=0\), then the series may also be called the Maclaurin series for \(f(x)\): \[\begin{aligned} Tf(x) &=\sum_{n=0}^\infty \dfrac{f^{(n)}(0)}{n!}x^n \\ &=f(0)+f'(0)x+\dfrac{f''(0)}{2}x^2 \\[2 pt] &\quad+\dfrac{f^{(3)}(0)}{3!}x^3+\cdots+\dfrac{f^{(n)}(0)}{n!}x^n+\cdots \end{aligned}\]

If the function \(f(x)\) is the limit of a power series centered at \(x=0\) then the series is its Maclaurin series: \[ f(x)=Tf(x) \]

Caution: The Taylor Series Theorem does not say that the Taylor series for \(f(x)\) actually converges to \(f(x)\). It merely says that if a power series converges to \(f(x)\), it must be its Taylor series, \(Tf(x)\).

In fact, there are functions, \(f(x)\), whose Taylor series, \(Tf(x)\), does not converge back to \(f(x)\). However, suffice it to say that for every function discussed in this course, the Taylor series does in fact converge to the function.

We will delay the discussion of when a Taylor series converges to the function until the end of this chapter when we discuss the Taylor Series Convergence Proofs. This is why we have notation to distinguish between the function, \(f(x)\), and its Taylor series, \(Tf(x)\). Then, for example, we can ask the question "Does \(Tf(x)\) converge to \(f(x)\)?" However, throughout most of this chapter, we assume the Taylor series does converge to the function. So we write \(f(x)\) and \(Tf(x)\) interchangeably.