Editing Taylor series (section)

==Definition==
The Taylor series of a [[real-valued function|real]] or [[complex-valued function]] {{math|''f''&thinsp;(''x'')}}, that is [[infinitely differentiable function|infinitely differentiable]] at a [[real number|real]] or [[complex number]] {{math|''a''}}, is the [[power series]]
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<math display="block"> f(a) + \frac {f'(a)}{1!}(x-a) + \frac{f''(a)}{2!} (x-a)^2+ \cdots = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} (x-a)^{n}. </math>
Here, {{math|''n''!}} denotes the [[factorial]] of {{mvar|n}}. The function {{math|''f''{{isup|(''n'')}}(''a'')}} denotes the {{mvar|n}}th [[derivative]] of {{mvar|f}} evaluated at the point {{mvar|a}}. The derivative of order zero of {{mvar|f}} is defined to be {{mvar|f}} itself and {{math|(''x'' − ''a'')<sup>0</sup>}} and {{math|0!}} [[empty product|are both defined to be&nbsp;1]]. This series can be written by using [[sigma notation]], as in the right side formula.{{sfn|Banner|2007|p=[https://books.google.com/books?id=OrumDwAAQBAJ&pg=PA530 530]}} With {{math|''a'' {{=}} 0}}, the Maclaurin series takes the form:{{sfn|Thomas|Finney|1996|loc=See §8.9.}}
<math display="block"> f(0)+\frac {f'(0)}{1!} x+ \frac{f''(0)}{2!} x^2+ \cdots = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} x^{n}. </math>