Maclaurin Series
Before working any illustrations of Taylor Series the first requirement is to address the assumption that a Taylor Series will in fact exist for a specified function. Let us start out with a few notation and definitions that we'll require.
To find out a condition that must be true in order for a Taylor series to exist for a function let's first describe the n^{th} degree Taylor polynomial of f (x) as,
.
Note: This actually is a polynomial of degree at most n! If we were to write out the sum with no the summation notation this would obviously be an n^{th} degree polynomial.