Can someone help me to find the closed form of this series: $\sum_{k=1}^\infty \frac{(-1)^k B_{k+1}}{k+1}$? I need to find a closed form for the following series where $B_{k+1}$ is the (k+1)th Bernoulli number. 
$$\sum_{k=1}^\infty \frac{(-1)^k B_{k+1}}{k+1}$$

Odd Bernoulli numbers are $0$ aside from $B_1$, so we have
$$\sum_{k=1}^\infty (-1)^k \frac{B_{k+1}}{k+1} = -\sum_{k=1}^\infty \frac{B_{2k}}{2k}$$
but $B_{2n} \sim (-1)^{n-1} 4 \sqrt{\pi n} \left( \frac{n}{\pi e} \right)^{2n}$, so your series won't converge.
