Understanding and Solving the Taylor Series for a Specific Point

[cytochrome]

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!

 

[jbunniii]

A Taylor series of a function f, about the point x = 0 is a representation of f as a sum of polynomials centered at x = 0, namely x, x^2, x^3, x^4, \ldots

Thus it will have the form

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

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

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

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

a_k = \frac{f^{(k)}(c)}{k!}

where f^{(0)} is taken to mean f, and f^{(k)} means the k'th derivative of f. So to calculate a_k, you find the k'th derivative of f and evaluate it at x = c.

 

[jbunniii]

As a simple example, take f(x) = e^x. This is a nice example because all of the derivatives also equal e^x. Thus, if I center the Taylor series at c, the coefficients will be
a_k = \frac{f^{(k)}(c)}{k!} = \frac{e^c}{k!}
and the series will therefore be
e^x = \frac{e^c}{0!} + \frac{e^c}{1!}(x - c) + \frac{e^c}{2!}(x - c)^2 + \ldots
For the special case c = 0, we have e^c = e^0 = 1 so the coefficient simplifies to
a_k = \frac{1}{k!}
and the series will therefore be
e^x = \frac{1}{0!} + \frac{1}{1!}x + \frac{1}{2!}x^2 + \ldots

 

[Simon Bridge]

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:

... 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.