In my calculus class, my teacher said that if one was to try to calculate the maclaurin or taylor series of $\tan x$ by strictly using the definition , then you would run into many problems and your answer would involve 'Bernoulli' numbers. That is why if we are to do the expansion for $\tan x$ , we must use division of power series.
But my question is, what are these numbers? Is it a just the ratio that appears when doing division of the series of $\sin x$ over $\cos x$? Is it some special ratio related to factorials?
Apologize if it is trivial, I have looked at other sources but I don't seem to fully understand this. Thank you all.
Bernoulli numbers are used in the evaluation of hyperbolic tangent and tangent functions. It is a analytical continuation of the sum of the infinite series
$\frac{1}{n^{s}}$
where the real part of the sum converges real part of $s>{1}$.
Something also which could help is looking at the Riemann Zeta function if you have not seen that.