Taylor Series

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 sums, 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.