Convergence of alternating series based on prime numbers I've been experimenting with some infinite series, and I've been looking at this one,
$$\sum_{k=1}^\infty (-1)^{k+1} {1\over p_k}$$
where $p_k$ is the k-th prime. I've summed up the first 35 terms myself and got a value of about 0.27935, and this doesn't seem close to a relation of any 'special' constants, except maybe $\frac12\gamma $.
My question is, has the sum of this series been proven to have a particular closed form? If so, what is this value?

As mentioned, this series has an expansion given by the OEIS.  This series is mentioned in many sources, such as Mathworld, Wells, Robinson & Potter and Weisstein.
These sources all seem to imply that, though the series converges, no known "closed form" for this sum exists.
