< 1 2 >
Hyperbolic sine
The power series for the hyperbolic sine is
for all values of x
Explanation
As the hyperbolic sine can be differentiated continuously, you get
The point x = 0 gives sinh (0) = 0 and cosh (0) = 1 so you get
We substitute this in the Taylor series and find
so
Symmetry
The formula only contains odd numbers, so there is the symmetry
General format
You can write the power series as a sum
Example 1
You can see that sinh (0) = 0, because
sinh (0) = 0 + 0 + 0 + ... = 0