Understanding Taylor Series Approximations

In summary, the Taylor series is a series of polynomial approximations of a function. It is most useful for functions that arenicely differentiable, but can be approximated by a simpler series if only information about the function at a single point is known.
  • #1
j-lee00
95
0
When it says "about a point x=a", what does this mean? why not just say at x = a?

Thanks
 
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  • #2
Because x is a variable. Saying that the Taylor series is "about the point x= a" means its interval of convergence is centered on a:
[tex]\sum a_n (x- a)^n[/tex].
 
  • #3
Because one is looking at a neighbourhood of a, say all x that satisfy |x-a|<d for some (small) number delta>0.
 
  • #4
why can be written taylor series?
 
  • #5
Despite the question mark, that is not a question.
 
  • #6
Landau said:
Despite the question mark, that is not a question.

can we ask like that: how do the taylor's series work?
Thanks
 
  • #7
Landau said:
Because one is looking at a neighbourhood of a, say all x that satisfy |x-a|<d for some (small) number delta>0.

HallsofIvy said:
Because x is a variable. Saying that the Taylor series is "about the point x= a" means its interval of convergence is centered on a:
[tex]\sum a_n (x- a)^n[/tex].

Thanks
 
  • #8
alpagut said:
can we ask like that: how do the taylor's series work?
Thanks
One way to look at it is this- how can we best approximate a function, given information about it at a single point?

If the only thing we know is that f(a)= A, then the simplest thing to do is to approximate f(x) by the constant A- and there is no reason to think that any more complicated formula would give a better approximation.

If we know that f(a)= A and f'(a)= B, then we can approximate f by the linear function satisying those properties: y= A+ B(x- a).

If we know that f(a)= A, f'(a)= B, and f"(b)= C, the simplest function having those properties is [itex]y= A+ B(x- a)+ (C/2)(x- a)^2[/itex].

Continuing in that way, gives the succesive "Taylor's polynomials". For especially "nice" functions, we can extend that to an infinite power series, the "Taylor's series".

(But be careful, even if a function is infinitely differentiable, so that we can form the "Taylor's series", it can happen that the Taylor's series does not converge to the function at more than single point.)
 
  • #9
HallsofIvy said:
One way to look at it is this- how can we best approximate a function, given information about it at a single point?

If the only thing we know is that f(a)= A, then the simplest thing to do is to approximate f(x) by the constant A- and there is no reason to think that any more complicated formula would give a better approximation.

If we know that f(a)= A and f'(a)= B, then we can approximate f by the linear function satisying those properties: y= A+ B(x- a).

If we know that f(a)= A, f'(a)= B, and f"(b)= C, the simplest function having those properties is [itex]y= A+ B(x- a)+ (C/2)(x- a)^2[/itex].

Continuing in that way, gives the succesive "Taylor's polynomials". For especially "nice" functions, we can extend that to an infinite power series, the "Taylor's series".

(But be careful, even if a function is infinitely differentiable, so that we can form the "Taylor's series", it can happen that the Taylor's series does not converge to the function at more than single point.)

Thank you!
 

1. 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 the behavior of a function near a specific point.

2. How is a Taylor series calculated?

A Taylor series is calculated using the derivatives of a function at a specific point. The coefficients of each term in the series are determined by plugging in the values of the derivatives into a specific formula.

3. What is the purpose of a Taylor series?

The purpose of a Taylor series is to approximate the behavior of a function near a specific point. It can be used to calculate the value of a function at a point, as well as to approximate the behavior of the function at nearby points.

4. What is the difference between a Taylor series and a Maclaurin series?

A Maclaurin series is a special case of a Taylor series where the point of approximation is set to 0. This means that the Maclaurin series only uses derivatives evaluated at 0, while a Taylor series can use derivatives evaluated at any point.

5. What are some real-world applications of Taylor series?

Taylor series are used in various fields of science and engineering, such as physics, economics, and computer science. They are used to approximate the behavior of physical systems, to optimize functions, and to develop algorithms for solving complex problems.

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