Difference between revisions of "Taylor Series"

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(New Concept: Taylor Series)
(New Concept: Taylor Series)
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:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
 
:<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
  
* Note that for nth approximate partial sums, the only information used about the function f, is its derivatives at a up to order n.
+
* Note that for nth approximate partial sum, the only information used about the function f, is its derivatives at a up to order n.
 
* '''Convergence:'''  A function whose Taylor series converges to the function is called ''analytic''.  But sometimes, as we have seen with power series generally, convergence can occur only within a radius of convergence.
 
* '''Convergence:'''  A function whose Taylor series converges to the function is called ''analytic''.  But sometimes, as we have seen with power series generally, convergence can occur only within a radius of convergence.
 
* Example: The Taylor series for cos(x), at a = 0.  Partial sums for 4 and 6 terms.
 
* Example: The Taylor series for cos(x), at a = 0.  Partial sums for 4 and 6 terms.

Revision as of 23:21, 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, verified, and adapted from Wikipedia, this page (permanent link). See license to copy, modify, distribute.

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}
  • Note that for nth approximate partial sum, the only information used about the function f, is its derivatives at a up to order n.
  • Convergence: A function whose Taylor series converges to the function is called analytic. But sometimes, as we have seen with power series generally, convergence can occur only within a radius of convergence.
  • Example: The Taylor series for cos(x), at a = 0. Partial sums for 4 and 6 terms.
cos(x) and Taylor series partial sums for cos(x), 4 and 6 terms

Try it yourself

Open Maxima and type (or copy and paste):

a:0;
n1:4;
n2:6;
M:4;
plot2d([taylor(cos(x),x,a,n1), taylor(cos(x),x,a,n2), cos(x)],[x,-M,M]);

After you have typed in the above you can change the parameters a, n1, n2, M by scrolling up and changing them, then pressing return.