Taylor Series and Maclaurin Series Doubt

[sarvesh0303]

Homework Statement

If I take a function f(x) and its taylor series, then will the infinite series give me the value of the function at any x value or will it only give proper values for x≈a?

For example, If I take a maclaurin series for a function will it give me proper values for all x values or only if x≈a=0?

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

If I take a function f(x) and its taylor series, then will the infinite series give me the value of the function at any x value or will it only give proper values for x≈a?

For example, If I take a maclaurin series for a function will it give me proper values for all x values or only if x≈a=0?

It always works for x = a. It is possible that is the only value for which it works. But more usually there is a radius of convergence where it works for |x-a|<r and the series diverges for |x-a|>r. Or it may converge and equal the function for all x. Generally, the farther you are from x = a, the more terms you need for a given accuracy. There are examples where the series converges for all x but doesn't equal the function except at x = a.

 

[HallsofIvy]

There exist functions for which the Taylor series exist and converges for all x but converges to the value of the function only at the given point. The function defined as $f(x)= e^{-1/x^2}$ for x not equal to 0, 0 for x= 0, has derivatives of all order and they are all 0 at x= 0 so the Maclaurin series (Taylor series with a= 0) is just 0 for all x. But f is 0 only at x= 0.

(Functions that have the property that they are equal to their Taylor series at every point on a set are called 'anlytic' on that set. Those are especially important in functions of complex numbers.)

 

[sarvesh0303]

So how could we find out whether the sum will give me the approximate value of the function or not?

 

[LCKurtz]

So how could we find out whether the sum will give me the approximate value of the function or not?

Take a look at:

http://en.wikipedia.org/wiki/Taylor's_theorem