Let $f(x) = \begin{cases}\ln\frac{x}{e^x-1}, \quad x > 0; \\ 0, \quad\qquad x=0; \\ \ln\frac{-x}{e^x-1}, \quad x < 0. \end{cases}$

Power series in 0: $f(x) = \sum_{n=1}^{\infty} a_n x^n = -\frac{x}{2} - \frac{x^2}{24} + \frac{x^4}{2880} + \ldots$

I am interested in estimates (asymptotic) for the sum of the coefficients $B(N) = \sum_{n=1}^{N} a_n$. For example, $|\ln \frac{1}{e-1} - B(N)| \ll \frac{1}{N}$.

Do you know of any references on this?

Thanks!