Taylor Series

To accommodate the center, we rewrite

displaymath89

Next we use the geometric series with tex2html_wrap_inline93 :

The series will converge for -1<x<11.

Taylor series and differentiation

It is easy to take derivatives of Taylor series: Just take the derivative term-by-term. The radius of convergence of the derivative will be the same as that of the original series.

This can be exploited to find Taylor series! Consider the example tex2html_wrap_inline103 . Its Taylor series has the form

We can then find the Taylor series with center of its derivative:

Since the constant term has derivative 0, the summation starts at n=1. We then just take the derivative of the general term:

N.B. Dividing both sides by -2x yields the maybe more interesting formula

Try it yourself!

Find the Taylor series with center

for the hyperbolic cosine function

by using the fact that

is the derivative of the hyperbolic sine function

, which has as its Taylor series expansion

displaymath111

(If you remember the Taylor expansions for and , you get an indication, why their hyperbolic counterparts might deserve the names "sine" and "cosine". )

Click here for the answer, or to continue.

Helmut Knaust
Tue Jul 16 11:25:05 MDT 1996