mfb said:If you want to approximate the function value somewhere, it can be useful to consider an expansion point close to that.
No - the function that is represented by the Taylor series has to be defined and infinitely differentiable at that point. So you can't write a Taylor series centered around a = 0 for ln(x).Turion said:Why? Can't you use any Taylor series centered at any point?
The idea is that if you know the exact value of your function at some point a, you can use a Taylor series to find approximations at points near a. For example, we know that cos(60°) = cos(##\pi/3##) = .5, exactly. We can use a Taylor series expansion about a = ##\pi/3## to get reasonable approximations to angles such as 60.5°, and so on.Turion said:Also, when I calculate e^2 using Taylor series centered at 2, I get 0 since x-a=2-2=0.
I just noticed that in order to calculate any power of e, you need to know that power of e first. Doesn't that make it kind of pointless?
I would like to point out that the function [itex]f(x)= e^{-1/x^2}[/itex] if [itex]x\ne 0[/itex], f(0)= 0, is infinitely differentiable so you can write a Taylor series centered around x= 0, just as Mark44 says but that Taylor series is not equal to f(x) anywhere except at x= 0.Mark44 said:No - the function that is represented by the Taylor series has to be defined and infinitely differentiable at that point. So you can't write a Taylor series centered around a = 0 for ln(x).
The idea is that if you know the exact value of your function at some point a, you can use a Taylor series to find approximations at points near a. For example, we know that cos(60°) = cos(##\pi/3##) = .5, exactly. We can use a Taylor series expansion about a = ##\pi/3## to get reasonable approximations to angles such as 60.5°, and so on.
Turion said:over a Maclaurin series?
Also, how do I calculate e^0 using Maclaurin series? I'm getting 0^0.
A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function at a specific point by using information about the function's derivatives at that point.
A Taylor series is centered at a nonzero value when the point of approximation is not at zero, but at a different value on the function's domain. This means that the series will be a representation of the function at that specific point.
Yes, there are benefits to using Taylor series centered at nonzero values. This allows for a more accurate approximation of a function at a specific point, especially when the function is not symmetric around the origin.
In theory, a Taylor series can be used for any function. However, the series may only converge for certain functions and may require an infinite number of terms to accurately represent the function.
The accuracy of a Taylor series centered at a nonzero value can be determined by comparing the series to the actual function at the point of approximation. The more terms that are used in the series, the more accurate the approximation will be.