To accommodate the center, we rewrite

Next we use the geometric series with :

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

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 -2*x* yields the maybe more interesting formula

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

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Tue Jul 16 11:25:05 MDT 1996

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