Difference between revisions of "Taylor Series"

Revision as of 23:11, 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.

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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}

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

@@ Line 27: / Line 27: @@
 :<math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math>
-* '''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, or not at all.
+* '''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.
 [[Image:taylorcos46.png|500px|thumb|center|cos(x) and Taylor series partial sums for cos(x), 4 and 6 terms ]]