Taylor Series

Review Concepts

• Sequences
• Convergence of infinite sequences
• 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 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.

Let f be a function with derivatives of all orders throughout some interval containing a as an interior point. 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.

• $a$ is where the derivatives are taken (center)
• n1 and n2 are the orders for the two taylor series plots
• [-M, M] is the range of the plot

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?

• What is imprecise about the previous question? Why should it make a mathematician cringe? Can you fix the problem?

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]);

• Is there an interval where the Taylor series seems to converge? What is it?

• Use the formula above to compute (by hand) the Taylor series to order 4 (i.e. n=4) of $4 x^2 + 2 x + 5$ at the point a=1.

Looking Ahead

• Taylor's theorem -- a precise statement about the error in a Taylor approximation of f(x) (ie. a partial sum of the Taylor series of f(x)).