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:
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Logarithmics
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has continuous derivatives up to (n+1)th order, then this function can be expanded in the following fashion:
, called the remainder after n+1 terms, is given by:
, that is,
, then the expansion is called the Taylor Series of 
.
the series is called the MacLaurin Series:
