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Taylor Series

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Degree \(n\) Polynomial

A polynomial in a variable \(x\) can always be written (or rewritten) in the form \[ a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0} \] where \(a_{i}\) (\(0 \le i \le n\)) are constants.

Using the summation notation, we can express the polynomial concisely by \[ \sum_{k=0}^{n} a_k x^k. \]

The highest degree in the above polynomial is \(n\), so the polynomial is called an \(n\) degree polynomial.

Degree \(n\) Polynomial as a Linear Combination of Monomials

A monomial in a variable \(x\) is a power of \(x\) where the exponent is a nonnegative integer (i.e. \(x^n\) where \(n\) is a nonnegative integer). You might see another definition of monomial which allows a nonzero constant as a coefficient in the monomial (i.e. \(a x^n\) where \(a\) is nonzero and \(n\) is a nonnegative integer). Then an \(n\) degree polynomial \[ \sum_{k=0}^{n} a_k x^k \] can be seen as a linear combination of monomials \(\{x^i |0 \le i \le n\}\).

Taylor Series Expansion, Infinite

A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The Taylor series expansion about \(x=x_0\) of a function \(f(x)\) that is infinitely differentiable at \(x_0\) is the power series \[ f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{f'''(x_0)}{3!}(x-x_0)^3+\dotsb \]

Using the summation notation, we can express the Taylor series concisely by \[ \sum_{k=0}^{\infty} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k .\]

Taylor Series Expansion, Finite

In practice, however, we often cannot compute the (infinite) Taylor series of the function, or the function is not infinitely differentiable at some points. Therefore, we often have to truncate the Taylor series (use a finite number of terms) to approximate the function.

Taylor Series Approximation of Degree \(n\)

If we use the first \(n+1\) terms of the Taylor series, we will get \[ T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k ,\] which is called a degree-n Taylor polynomial.

Error Bound on a Taylor Series Expansion of Degree \(n\)

Suppose that \(f(x)\) is an \(n+1\) times differentiable function of \(x\), and \(T_n(x)\) is the Taylor polynomial of degree \(n\) for \(f(x)\) centered at \(x_0\). Then when \(h = x-x_0 \to 0\), we can get the error bound by \[ f(x)-T_n(x)\le C \cdot h^{n+1} = O(h^{n+1}) \]

We will see the exact expression of \(C\) in the next section: Taylor Remainder Theorem.

Taylor Remainder Theorem

Suppose that \(f(x)\) is an \(n+1\) times differentiable function of \(x\). Let \(R_n(x)\) denote the difference between \(f(x)\) and the Taylor polynomial of degree \(n\) for \(f(x)\) centered at \(x_0\). Then \[ R_n(x) = f(x) - T_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-x_0)^{n+1} \] for some \(\xi \in [x_0, x]\).

Example of a Taylor Series Expansion

Suppose we want to expand \(f(x) = \cos x\) about the point \(x_0 = 0\). Following the formula \[ f(x) = f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{f'''(x_0)}{3!}(x-x_0)^3+\dotsb, \] we need to compute the derivatives of \(f(x) = \cos x\) at \(x = x_0\). \[\begin{align*} f(x_0) &= \cos(0) = 1\\ f'(x_0) &= -\sin(0) = 0\\ f''(x_0) &= -\cos(0) = -1\\ f'''(x_0) &= \sin(0) = 0\\ f^{(4)}(x_0) &= \cos(0) = 1\\ &\vdots \end{align*}\] Then \[\begin{align*} \cos x &= f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\dotsb\\ &= 1 + 0 - \frac{1}{2}x^2 + 0 +\dotsb\\ &= \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!}x^{2k} \end{align*}\]

Example of Using a Truncated Taylor Series to Approximate a Function

Suppose we want to approximate \(f(x) = \sin x\) at \(x = 2\) using a degree-4 Taylor polynomial about (centered at) the point \(x_0 = 0\). Following the formula \[ f(x) \approx f(x_0)+\frac{f'(x_0)}{1!}(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\frac{f'''(x_0)}{3!}(x-x_0)^3+\frac{f^{(4)}(x_0)}{4!}(x-x_0)^4, \] we need to compute the first \(4\) derivatives of \(f(x) = \sin x\) at \(x = x_0\). \[\begin{align*} f(x_0) &= \sin(0) = 0\\ f'(x_0) &= \cos(0) = 1\\ f''(x_0) &= -\sin(0) = 0\\ f'''(x_0) &= -\cos(0) = -1\\ f^{(4)}(x_0) &= \sin(0) = 0 \end{align*}\] Then \[\begin{align*} \sin x &\approx f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\frac{f^{(4)}(0)}{4!}x^4\\ &= 0 + x + 0 - \frac{1}{3!}x^3 + 0 \\ &= x - \frac{1}{6}x^3 \end{align*}\] Using this truncated Taylor series centered at \(x_0 = 0\), we can approximate \(f(x) = \sin(x)\) at \(x=2\). To do so, we simply plug \(x = 2\) into the above formula for the degree 4 Taylor polynomial giving \[\begin{align*} \sin(2) &\approx 2 - \frac{1}{6} 2^3 \\ &\approx 2 - \frac{8}{6} \\ &\approx \frac{2}{3} \end{align*}.\]

Example of an Error Bound

Suppose we want to approximate \(f(x) = \sin x\) using a degree-4 Taylor polynomial expanded about the point \(x_0 = 0\). We want to compute the error bound for this approximation. Following Taylor Remainder Theorem, \[ R_4(x) = \frac{f^{(5)}(\xi)}{5!} (x-x_0)^{5} \] for some \(\xi \in [x_0, x]\).

If we want to find the upper bound for the absolute error, we are looking for an upper bound for \(|f^{(5)}(\xi)|\).

Since \(f^{(5)}(x) = \cos x\), we have \(|f^{(5)}(\xi)|\le 1\). Then \[\begin{align*} |R_4(x)| &= \left|\frac{f^{(5)}(\xi)}{5!} (x-x_0)^{5}\right| \\ &\le \frac{|f^{(5)}(\xi)|}{5!} |x|^{5} \\ &= \frac{1}{120} |x|^{5} \end{align*}\]

Finite Difference Approximation

For a differentiable function \(f:\mathbb{R} \rightarrow \mathbb{R}\), the derivative is defined as

\[f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \]

Let's consider the finite difference approximation to the first derivative as

\[f'(x) \approx \frac{f(x+h)-f(x)}{h} \]

where \(h\) is often called a "perturbation", i.e., a "small" change to the variable \(x\) (small when compared to the magnitude of \(x\)). By the Taylor's theorem, we can write

\[f(x+h) = f(x) + f'(x)\, h + f''(\xi)\, \frac{h^2}{2} \]

for some \(\xi \in [x,x+h]\). Rearranging the above we get

\[f'(x) = \frac{f(x+h)-f(x)}{h} - f''(\xi)\, \frac{h}{2} \]

Therefore, the truncation error of the finite difference approximation is bounded by \(M\,h/2\), where \(M\) is a bound on \(|f''(\xi)|\) for \(\xi\) near \(x\).

Reference text: "Scientific Computing: an introductory survey" by Michael Heath

Review Questions

  1. What is the general form of a Taylor series?
  2. How do you use a Taylor series to approximate a function at a given point?
  3. How can you approximate the derivative of a function using Taylor series?
  4. How can you approximate the integral of a function using Taylor series?
  5. Given an function and a cneter, can you write out the \(n\)-th degree Taylor polynomial?
  6. For an \(n\)-th degree Taylor polynomial, what is the bound on the error of your approximation as a function of distance from the center?
  7. For simple functions, can you find the constant \(C\) in the Taylor error bound?
  8. Be able to determine how many terms are required for a Taylor series approximation to have less than some given error.

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