Difference between revisions of "Taylor Series"

Revision as of 20:45, 14 March 2011

Think of Taylor series as a special kind of power series, where the sequence of partial sums are meant as better and better approximations of some other function.
The Taylor Series is derived from the function.

Definition from

The Taylor series of a function ƒ(x) at a is the power series

f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots.

which can be written in the more compact sigma notation as

\sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}

@@ Line 23: / Line 23: @@
 -->
-which can be written in the more compact [[sigma notation]] as
+which can be written in the more compact sigma notation as
 :<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
-where ''n''! denotes the [[factorial]] of ''n'' and ''ƒ''<sup>&nbsp;(''n'')</sup>(''a'') denotes the ''n''th [[derivative]] of ''ƒ'' evaluated at the point ''a''. The zeroth derivative of ''ƒ'' is defined to be ''ƒ'' itself and {{nowrap|(''x'' &minus; ''a'')<sup>0</sup>}} and 0! are both defined to be&nbsp;1. In the case that {{nowrap|''a'' {{=}} 0}}, the series is also called a '''Maclaurin series'''.