A few questions about the Taylor series

[Shing]

When I tried to learn the Taylor series , I could not comprehend why a infinite series can represent a function

Would anyone be kind enough to teach me the Taylor series? thank you

the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

$$\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$$

PS. I am 18 , having the high school Math knowledge including Calculus

 

[EnumaElish]

Taylor series is a special case of the polynomial approximation theorem:

http://ccrma.stanford.edu/~jos/mdft/Weierstrass_Approximation_Theorem.html

 

[Shing]

Taylor series is a special case of the polynomial approximation theorem:

http://ccrma.stanford.edu/~jos/mdft/Weierstrass_Approximation_Theorem.html

thank you for your help
But I don't really got it..
A function has its own domain and range... but what is the domain and range of a series?

 

[mrandersdk]

you can also consider a serie as a function

$$f(x) = \sum_0^{\infty} \frac{f^{(n)(a)}}{n!}(x-a)^n$$

if the sum converges this is nothing but a function, to each x you get a number, namely the sum of the series.

Mayby an example is in place.

Lets say we define a function by

$$f(x) = \sum_0^{\infty} (g(x))^n$$

where $$g: \mathhb{R} \rightarrow ]0,1[$$, is this series a function you could say. But because the range of g i ]0,1[ this is just the geometric series, that is

$$f(x) = \sum_0^{\infty} (g(x))^n = \frac{1}{1-g(x)}$$

so this is indeed a function. So you see that a series is a function if the sum converges. It's although not always possible to find a so simple expression like here for the serie.

 

[mrandersdk]

a little text from wikipedia

The Taylor series need not in general be a convergent series, but often it is. The limit of a convergent Taylor series need not in general be equal to the function value f(x), but often it is. If f(x) is equal to its Taylor series in a neighborhood of a, it is said to be analytic in this neighborhood. If f(x) is equal to its Taylor series everywhere it is called entire.

 

[HallsofIvy]

thank you for your help
But I don't really got it..
A function has its own domain and range... but what is the domain and range of a series?

?? If a series converges to a function, then it's domain and range are exactly that of the function, of course. A series, depending on a variable x, is a function. You seem to think there is something special about using a series to define a function.

 

[mrandersdk]

?? If a series converges to a function, then it's domain and range are exactly that of the function, of course. A series, depending on a variable x, is a function. You seem to think there is something special about using a series to define a function.

you are right, but sometimes the taylor series only converges in a neigborhood around the point a, and sometimes it doesn't converges at all. So to say that if the series converges then it is equal to the function is a bit loose, I agree on that for those x where the series converges, then on those x they agree. Actually that is probably what you meen, but just to make it clear.

 

[HallsofIvy]

I didn't say that! I said that if a series converges, it is necessarily equal to a function. I didn't say anything about the series being a Taylor's series.