Difference between revisions of "Taylor Series"

Revision as of 23:14, 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.

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}

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

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])

@@ Line 31: / Line 31: @@
 * 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 ]]
+== 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])