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Deriving the Taylor Polynomial

This note has been used to help create the Taylor Series wiki

Suppose we want to interpolate an infinite number of points on the Cartesian plane using a continuous and differentiable function \(f\). How can this be done?

Solution

Given \(n\) points on the Cartesian plane, the set of points can be interpolated using a polynomial of at least degree \(n-1\). Given an infinite number of points to interpolate, we need an infinite polynomial:

\[f(x) = {a}_{0} + {a}_{1}(x-{x}_{0}) + {a}_{2}{(x-{x}_{0})}^{2} +...\] where \(\left|x-{x}_{0}\right|\) is within the radius of convergence.

Observation: \[f({x}_{0}) = {a}_{0}\] \[f'({x}_{0}) = {a}_{1}\] \[f''({x}_{0}) = 2{a}_{2}\] \[f'''({x}_{0}) = 6{a}_{3}\] \[{f}^{(4)}({x}_{0}) = 24{a}_{4}\] \[{f}^{(n)}({x}_{0}) = n!{a}_{n}\]

Solving for each constant term expands the original function into the infinite polynomial: \[f(x) = \sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } { f }^{ (n) }({ x }_{ 0 } } ){ (x-{ x }_{ 0 }) }^{ n }.\]

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
2 years, 5 months ago

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I find this interesting. I use Taylor Series a lot but I had never thought of what actually gives rise to them. Great note! Ethan Robinett · 2 years, 5 months ago

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@Ethan Robinett There is probably a more rigorous proof out there. This note is more of an intuitive derivation than a proof. Steven Zheng · 2 years, 5 months ago

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@Steven Zheng I still thought it was pretty informative. Ethan Robinett · 2 years, 5 months ago

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@Ethan Robinett Well, calculus was far from rigorous during Taylor's time. Steven Zheng · 2 years, 5 months ago

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@Steven Zheng It's actually interesting to think about that. It was like at one point in history, someone decided that math in general needed to be more rigorous. Up until that point, everyone was just kinda throwing around ideas without too much proof. Ethan Robinett · 2 years, 5 months ago

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@Ethan Robinett Throwing around ideas without too much proof is how progress is made. I think, in the early stage of development, mathematical ideas were discovered to work. Well-polished theories come later when exceptions are found. This often happens when there are new ways of looking at things. Steven Zheng · 2 years, 5 months ago

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