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

[tex] f(x) = \sum_0^{\infty} (g(x))^n [/tex]

where [tex] g: \mathhb{R} \rightarrow ]0,1[ [/tex], 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

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.

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.

?? 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 probaly what you meen, but just to make it clear.

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.