# A few questions about the Taylor series

• Shing
In summary, 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. It is a special case of the polynomial approximation theorem and can be considered as a function if it converges. However, the limit of a convergent Taylor series may not always be equal to the function value, and the series may only converge in a certain neighborhood around the point.
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

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.

Last edited:
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.

Shing said:
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.

HallsofIvy said:
?? 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.

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.

## 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 a function at a given point by using its derivatives.

## Why is the Taylor series important?

The Taylor series is important because it allows us to approximate complex functions with simpler ones, making it easier to calculate values and solve problems related to the function.

## What is the formula for a Taylor series?

The formula for a Taylor series is f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ..., where a is the point at which the function is being approximated and f'(a), f''(a), etc. are the derivatives of the function at that point.

## How do you find the coefficients of a Taylor series?

The coefficients of a Taylor series can be found by taking the derivatives of the function at the given point, evaluating them at that point, and dividing by the factorial of the corresponding term number.

## What is the difference between a Maclaurin series and a Taylor series?

A Maclaurin series is a special case of a Taylor series where a = 0. This means that the function is being approximated at the point x = 0. In general, a Taylor series can be centered around any point, while a Maclaurin series is only centered around x = 0.

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