# Taylor Series

To accommodate the center, we rewrite

Next we use the geometric series with :

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 . 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

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