Difference between revisions of "Taylor Series"

Revision as of 00:26, 15 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}

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

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.

Does the series seem to converge in the range [-M,M] =[-8,8]? If so, how large should n1 or n2 be so that this convergence is apparent?

What is imprecise about the previous question? Why should it make a mathematician cringe?

Instead of cos(x) try the function

$f(x) = \frac{1}{x^2+1}$

plot2d([taylor(1/(x^2+1),x,a,n1), taylor(1/(x^2+1),x,a,n2), 1/(x^2+1)],[x,-M,M]);

@@ Line 48: / Line 48: @@
 * [-M, M] is the range of the plot
-== Questions for Exploration in Groups of Three (With Computer) ==
+== Questions for Exploration and Discussion in Groups of Three (With Computer) ==
 * Does the series seem to converge in the range [-M,M] =[-8,8]?  If so, how large should n1 or n2 be so that this convergence is apparent?