Understanding the Derivation of Taylor Series

Thread starter gursimran
Start date Jun 9, 2011
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In summary, a Taylor series is a way to approximate a function by breaking it down into simpler functions. It is derived using Taylor expansion and involves finding derivatives and constructing a polynomial. Assumptions are made about the function's differentiability and convergence to the actual value. While theoretically applicable to any infinitely differentiable function, practical issues may arise with more complex functions.

Jun 9, 2011

gursimran

I read wikipedia article also but I can't find the proof of taylor series and from where it came from??

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Jun 9, 2011

micromass

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Hi gursimran!

Check out "home.iitk.ac.in/~psraj/mth101/lecture_notes/lecturer10.pdf"[/URL]
Taylor's theorem is 10.2

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Jun 9, 2011

gursimran

oh yeah!
thanks a lot..

Related to Understanding the Derivation of Taylor Series

1. What is the purpose of a Taylor series?

A Taylor series is a mathematical tool used to approximate a function by representing it as a sum of infinitely many simpler functions. It allows for the estimation of a function's behavior at a specific point or in a small interval around that point.

2. How is a Taylor series derived?

A Taylor series is derived using a process called Taylor expansion, which involves finding the derivatives of a function at a specific point and using those derivatives to construct a polynomial approximation of the function. This polynomial becomes the Taylor series for that function at that point.

3. What are the key steps in deriving a Taylor series?

The key steps in deriving a Taylor series include finding the derivatives of the function at a specific point, evaluating those derivatives at that point, and using those values to construct a polynomial. The polynomial is then simplified and written in the form of a Taylor series.

4. What are the assumptions made when deriving a Taylor series?

The main assumptions made when deriving a Taylor series are that the function is infinitely differentiable at the given point and that the series converges to the actual value of the function at that point. These assumptions may not hold for all functions and points, and may result in a Taylor series that does not accurately represent the function.

5. Can a Taylor series be used for any function?

In theory, a Taylor series can be used for any infinitely differentiable function. However, in practice, the series may not accurately represent the function for all points due to convergence issues. Additionally, calculating higher-order derivatives may become increasingly complicated for more complex functions.