A brief introduction to Taylor series

The Taylor series is a method of turning non ‘nice’ polynomial functions into polynomial functions. By nice I mean: trigonometric, exponential, logarithmic. Now, this all probably sounds too good to be true and you may scratch your head in disbelief and ask yourself “How could it be possible to do such a thing?” And I say to that, sometimes dreams can come true.

0. A deep relation between derivatives and coefficients of a polynomial

Consider a general polynomial p(x) of degree m, then it must be in the form