What is Taylor Series

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What is Taylor Series

Technical Definition: A way to approximate (very) smooth functions (for which derivatives up to high orders exist and are continuous)

Simple Definition: Taylor series is the infinite sum of all the terms for a specific function which is a very close approximation to the real value of the function.

Mathematical definition:

f(x) = f(a) + (f^'(a)(x-a))/1! + (f^''(x)(x-a)²)/2! + (f^'''(x)(x-a)³)/3! + ... +

Why do we have an infinite number of terms?:

The more terms we include in our approximated function, the better the approximation to the real value. For a graph this means that it will represent the actual graph function more.

Special Case (Maclaurin Series):

Maclaurin series is based of the Taylor Series, but we choose the function to be around origin (value = 0) rather than anywhere else.

Advantage of using Taylor/Maclaurin series

its allows for incredibly accurate approximations of a function (depending on the number of terms included)

Provide for integration and differentiation of functions to arrive at representations of other function

Disadvantage of using Taylor/Maclaurin series

some calculations become tedious or the series doesn’t converge quickly

many of the functions are limited to a certain domain given a specific range for convergence (some Taylor series are only valid for a small domain)

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