Tangent
The power series for the tangent is written as
Explanation
The derivatives of the tangent are
The point x = 0 gives tan (0) = 0, so
We substitute this in the Taylor series and find
so that
or written differently
General format
You can write the power series as a sum
where Un is the n-th up/down number. Or
where Bn is the n-th Bernoulli number.
Example 1
You can see that tan (0) = 0, because
tan (0) = 0 + 0 + 0 + ... = 0