If a function has continuous derivatives up to (n+1)^{th} order, then this function can be expanded in the following fashion:

where , called the remainder after n+1 terms, is given by:

When this expansion converges over a certain range of , that is, , then the expansion is called the Taylor Series of expanded about .

If the series is called the MacLaurin Series:

Logarithmics

Inverse Trigonometric Functions

Hyperbolic Functions

Inverse Hyperbolic Functions