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.
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)