Difference between revisions of "Taylor Series"

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(→‎New Concept: Taylor Series)
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* The Taylor Series is derived from the function.
 
* The Taylor Series is derived from the function.
  
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Definition adapted from Wikipedia,  
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Definition copied and adapted from Wikipedia, [http://en.wikipedia.org/w/index.php?title=Taylor_series&oldid=418169005 this page (permanent link)]
  
 
The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series
 
The Taylor series of a function ''ƒ''(''x'') at ''a'' is the power series

Revision as of 20:49, 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 copied and adapted from Wikipedia, this page (permanent link)

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}
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