Difference between revisions of "Taylor Series"
From Sean_Carver
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== New Concept: Taylor Series == | == New Concept: Taylor Series == | ||
| − | 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. | + | * 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. | ||
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| + | Definition from | ||
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| + | The Taylor series of a [[real number|real]] or [[complex number|complex]] function ''ƒ''(''x'') that is [[Infinitely differentiable function|infinitely differentiable]] in a [[Neighborhood (mathematics)|neighborhood]] of a [[real number|real]] or [[complex number]] ''a'' is the [[power series]] | ||
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| + | <!-- | ||
| + | As stated below, the Taylor series need not equal the function. So please don't write f(x)=... here. In other words, | ||
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| + | DO NOT CHANGE ANYTHING ABOUT THIS FORMULA-->:<math>f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. </math><!-- | ||
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| + | --> | ||
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| + | which can be written in the more compact [[sigma notation]] as | ||
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| + | :<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math> | ||
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| + | where ''n''! denotes the [[factorial]] of ''n'' and ''ƒ''<sup> (''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'' − ''a'')<sup>0</sup>}} and 0! are both defined to be 1. In the case that {{nowrap|''a'' {{=}} 0}}, the series is also called a '''Maclaurin series'''. | ||
Revision as of 20:43, 14 March 2011
Review Concepts
- Sequences
- Convergence
- Infinite series
- The sequence of partial sums of an infinite series
- Power series
New Concept: Taylor Series
- 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 real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a is the power series
which can be written in the more compact sigma notation as
where n! denotes the factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a. The zeroth derivative of ƒ is defined to be ƒ itself and Template:Nowrap and 0! are both defined to be 1. In the case that Template:Nowrap, the series is also called a Maclaurin series.
