Formula for the general term of the Taylor series of $\tan(x)$ at $x = 0$
Today we were taught different expansions; one of them was the series expansion of $\tan(x)$, $$\tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15} + \cdots .$$ So, with curiosity, I asked my sir about next term. He said to get general formula divide series expansion of $\sin x,\cos x$. His reply didn't satisfy me. Do we have a general term for this expansion in elementary functions? If not, then why?
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$\begingroup$We can write it using the Bernoulli numbers $B_n$: $$\tan x \sim \sum_{k = 1}^{\infty} \frac{(-1)^{k - 1} 4^k (4^k - 1) B_{2k}}{(2 k)!} x^{2 k - 1}.$$ The radius of convergence is $\frac{\pi}{2}$. (As one might guess, the series for $\tanh$ is the same, with the sign correction term $(-1)^{k - 1}$ removed.)
One can also produce terms in the expansion of this series using the well-known series for $\arctan$ and solving term-by-term.
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