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Calculus Textbooks Boundless Calculus Differential Equations, Parametric Equations, and Sequences and Series
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Calculus
Concept Version 7
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Applications of Taylor Series

Taylor series expansion can help approximating values of functions and evaluating definite integrals.

Learning Objective

  • Describe applications of the Taylor series expansion


Key Points

    • The partial sums of the series, which is called the Taylor polynomials, can be used as approximations of the entire function.
    • Differentiation and integration of power series can be performed term by term, and hence could be easier than directly working with the original function.
    • The (truncated) series can be used to compute function values numerically. This is particularly useful in evaluating special mathematical functions (such as Bessel function).

Terms

  • definite integral

    the integral of a function between an upper and lower limit

  • analytic function

    a real valued function which is uniquely defined through its derivatives at one point

  • complex analysis

    theory of functions of a complex variable; a branch of mathematical analysis that investigates functions of complex numbers


Full Text

Uses of the Taylor series for analytic functions include:

1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the entire function. These approximations are often good enough if sufficiently many terms are included. Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.

Taylor Polynomials

As more terms are added to the Taylor polynomial, it approaches the correct function. This image shows $\sin x$ and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.

2. Differentiation and integration of power series can be performed term by term and is hence particularly easy. Taylor series is especially useful in evaluating definite integrals. For example, to evaluate $\int_{0}^{1} e^{x^3} \, dx$, the Taylor series expansion of $e^t= \sum_{n=0}^{\infty} \frac{1}{n! } \, t^n$and the substitution of $t=x^3$ can be used. Since each term in the summation can be integrated separately, we can evaluate the definite integral as long as the sum converges.

3. An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available.

4. The (truncated) series can be used to compute function values numerically. This is particularly useful in evaluating special mathematical functions (such as Bessel function).

5. Algebraic operations can be done readily on the power series representation; for instance the Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. $e^{ix}$ can be found from the Taylor expansion of $\cos(x)$ and $\sin(x)$:

$\displaystyle{\cos(x) = 1-\frac{x^2}{2!}+\frac{x^4}{4! } -\frac{x^6}{6! }+ \cdots}$

$\displaystyle{\sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5! } -\frac{x^7}{7! } + \cdots}$

and adding the two terms together yields:

$\begin{aligned} cos(x)+i\,sin(x) & = (1-\frac{x^2}{2!}+\frac{x^4}{4! } - \cdots) + i (x - \frac{x^3}{3! } + \frac{x^5}{5! } - \cdots) \\ & = 1 + ix + \frac{(ix)^2}{2! } + \frac{(ix)^3}{3! }+ \frac{(ix)^4}{4! } + \cdots \\ & = e^{ix} \end{aligned}$

This result is of fundamental importance in many fields of mathematics (for example, in complex analysis), physics and engineering.

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