Can someone help me to find the closed form of this series: $\sum_{k=1}^\infty \frac{(-1)^k B_{k+1}}{k+1}$?

Question

I need to find a closed form for the following series where $B_{k+1}$ is the (k+1)th Bernoulli number.

$$\sum_{k=1}^\infty \frac{(-1)^k B_{k+1}}{k+1}$$

Answer 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.