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Power series for the inverse hyperbolic sine
The inverse hyperbolische sine can be expressed as a power series
Explanation
The first derivative of the inverse hyperbolic sine is
We determine the further derivatives
The point x = 0 gives
We substitute this in the Taylor series and find
Now we see a regularity, because
so that
The series is an odd function, because the exponents are all odd.
General format
You can write the series as a sum
Example 1
You can see that sinh-1(0) = 0, because
sinh-1(0) = 0 − 0 + 0 − 0 + 0 ... = 0