Power Series Expansion for Tangent Function/Sequence

Theorem
The Power Series Expansion for Tangent Function begins:


 * $\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$

Proof
From Power Series Expansion for Tangent Function:

Enumerating the Bernoulli numbers:

Thus the appropriate arithmetic is performed on each coefficient:

Hence the result.