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 -2x 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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