Does the sum $\sum_{k=1}^{\infty}B_{(4k-2)}+B_{(4k)}$ converges?

Question

Just as the title says I'd like to know if this sum $$ \sum_{k=1}^{\infty}(B_{(4k-2)}+B_{(4k)}) $$ converges and if so to which value. Here $B_{2k}$ are Bernoulli numbers.

I've tried with Mathematica but it seems inconclusive.

Thanks.

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EDIT: Meanwhile I found this related question: What is sum of the Bernoulli numbers? .

Answer 1

According to Wikipedia,

$$ |B_{2 n}|\sim4\sqrt{\pi n}\left(\frac n{\pi\mathrm e}\right)^{2n} $$

as $n\to\infty$. It follows that $B_{4k-2}+B_{4k}$ doesn't go to zero as $k\to\infty$, so the series diverges.