Talor series expansion of roots of algebraic equation

In summary, a Taylor series expansion is a mathematical tool used to approximate functions as an infinite sum of simpler functions. It can be used to find roots of algebraic equations by setting the polynomial function equal to zero and solving for the roots. However, it can only be applied to smooth, continuous functions and its accuracy depends on the order of the expansion and the smoothness of the function. There are also limitations to using a Taylor series expansion, such as difficulties with multiple roots or roots with high multiplicities.
  • #1
marellasunny
255
3
I have a algebraic equation like so:
x^2-1-εx=0

the roots are obviously-
x=ε/2±√(1+ε^2/4)

How can I expand the expression for the roots- as a taylor series?

the answer is given as:
x(1)=1+ε/2+ε^2/8+O(ε^3)

I am assuming the author expanded the root 'x' in terms of ε before hand and substituted in the algebraic.Is that even allowed?
 
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  • #2
Use the binomial theorem on √(1+ε^2/4) = (1 + ε2/4)1/2 = 1+ε2/8 + O(ε4)
 

1. What is a Taylor series expansion?

A Taylor series expansion is a mathematical tool used to approximate a function as an infinite sum of simpler functions. It is based on the idea that any smooth function can be approximated by a polynomial function.

2. How is a Taylor series expansion used to find roots of algebraic equations?

A Taylor series expansion can be used to approximate the roots of an algebraic equation by setting the polynomial function equal to zero and solving for the roots. This method is particularly useful for finding complex roots of equations.

3. Can a Taylor series expansion be used for any type of function?

No, a Taylor series expansion can only be used for smooth, continuous functions. It is not applicable for functions with discontinuities or sharp corners.

4. How accurate is a Taylor series expansion in finding roots of algebraic equations?

The accuracy of a Taylor series expansion depends on the order of the expansion and the smoothness of the function. Higher order expansions will generally provide more accurate results, but may also require more terms in the series.

5. Are there any limitations to using a Taylor series expansion for finding roots?

Yes, there are limitations to using a Taylor series expansion. It may not always converge to the exact root of an equation, and it may also encounter difficulties when dealing with multiple roots or roots with high multiplicities.

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