Understanding and Solving the Taylor Series for a Specific Point

In summary, the Taylor series is a representation of a function f as a sum of polynomials, centered at some point x=0. The k'th coefficient of the Taylor series is given by a_k = \frac{f^{(k)}(c)}{k!}.
  • #1
cytochrome
166
3
What does it mean to calculate the Taylor series ABOUT a particular point?

I understand the formula for the Taylor series but how do you solve it about a particular point for a function? It's the about the particular point that confuses me.

Could someone please explain this and provide examples?

Thanks!
 
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  • #2
A Taylor series of a function [itex]f[/itex], about the point [itex]x = 0[/itex] is a representation of [itex]f[/itex] as a sum of polynomials centered at [itex]x = 0[/itex], namely [itex]x, x^2, x^3, x^4, \ldots[/itex]

Thus it will have the form

[tex]f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + \ldots[/tex]

More generally, we can express [itex]f[/itex] as a sum of polynomials centered at some other point, say [itex]x = c[/itex]. Then it will look like

[tex]f(x) = a_0 + a_1 (x - c) + a_2(x - c)^2 + a_3 (x - c)^3 + a_4 (x - c)^4 + \ldots[/tex]

This is a more general form: if I set [itex]c = 0[/itex] then it reduces to the previous equation. Note that the values of the coefficients generally change if I change [itex]c[/itex]. Here is the formula for the k'th coefficient:

[tex]a_k = \frac{f^{(k)}(c)}{k!}[/tex]

where [itex]f^{(0)}[/itex] is taken to mean [itex]f[/itex], and [itex]f^{(k)}[/itex] means the k'th derivative of [itex]f[/itex]. So to calculate [itex]a_k[/itex], you find the k'th derivative of [itex]f[/itex] and evaluate it at [itex]x = c[/itex].
 
  • #3
As a simple example, take [itex]f(x) = e^x[/itex]. This is a nice example because all of the derivatives also equal [itex]e^x[/itex]. Thus, if I center the Taylor series at [itex]c[/itex], the coefficients will be
[tex]a_k = \frac{f^{(k)}(c)}{k!} = \frac{e^c}{k!}[/tex]
and the series will therefore be
[tex]e^x = \frac{e^c}{0!} + \frac{e^c}{1!}(x - c) + \frac{e^c}{2!}(x - c)^2 + \ldots[/tex]
For the special case [itex]c = 0[/itex], we have [itex]e^c = e^0 = 1[/itex] so the coefficient simplifies to
[tex]a_k = \frac{1}{k!}[/tex]
and the series will therefore be
[tex]e^x = \frac{1}{0!} + \frac{1}{1!}x + \frac{1}{2!}x^2 + \ldots[/tex]
 
  • #4
The Taylor series is only defined to be about a particular point. The Taylor series expansion of y(x) about point x=a would be: $$y(x)_a = \sum_{n=0}^\infty \frac{f^{(n)}}{n!}(x-a)^n$$ It will be exact at that point - see:

220px-Exp_series.gif


... this is the Taylor series expansion (red line), taken about x=0, for the exponential function. See how after each term is added in, the red line has the same value as the blue at x=0? As an exercise, try doing the expansion for ##y(x)=e^x## about ##x=2##, and compare.
 

What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function and is often used in calculus, numerical analysis, and other areas of mathematics.

How is a Taylor series calculated?

To calculate a Taylor series, one must first find the derivatives of the function at a certain point. Then, the derivatives are evaluated at that point and used to construct the terms of the series. The series is then summed to approximate the function at that point.

What is the purpose of a Taylor series?

The purpose of a Taylor series is to approximate a function and its behavior at a specific point. It can be used to simplify complex functions and make them easier to work with. It is also useful for numerical analysis and solving differential equations.

What are the applications of Taylor series?

Taylor series have many applications in mathematics and science. They are used in physics to model physical phenomena, in engineering to design efficient algorithms, and in economics to predict market trends. They are also used in computer graphics to create smooth curves and surfaces.

How accurate is a Taylor series?

The accuracy of a Taylor series depends on the number of terms used in the series. The more terms included, the more accurate the approximation will be. However, even with a large number of terms, a Taylor series may still have some errors, as it is an approximation and not an exact representation of the function.

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