Homework Help: Taylor Series and Maclaurin Series Doubt

1. Oct 9, 2012

sarvesh0303

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. The attempt at a solution

2. Oct 9, 2012

LCKurtz

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.

3. Oct 9, 2012

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.)

4. Oct 10, 2012

sarvesh0303

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

5. Oct 10, 2012

LCKurtz

Take a look at:

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