Taylor Series and Maclaurin Series Doubt

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Homework Help Overview

The discussion revolves around the Taylor series and Maclaurin series, specifically addressing their convergence and the conditions under which they accurately represent a function. Participants are exploring whether these series provide valid function values for all x or only near a specific point.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the conditions for convergence of Taylor series and whether they yield accurate function values outside a neighborhood of the center point. There are attempts to clarify the concept of radius of convergence and the behavior of specific functions.

Discussion Status

The discussion is active, with participants sharing insights about the convergence properties of Taylor series. Some have provided examples of functions where the series converges but does not equal the function outside a specific point, while others are seeking further clarification on how to determine the accuracy of the series representation.

Contextual Notes

Participants are considering functions that may have derivatives of all orders yet do not converge to the function value outside a certain range. There is also mention of external resources for further exploration of Taylor's theorem.

sarvesh0303
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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

 
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sarvesh0303 said:

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

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