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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?

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    http://forums.xkcd.com/viewtopic.php?f=17&t=59644&view=previous2012-11-21
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    More directly, see "[Prime sums](http://mathworld.wolfram.com/PrimeSums.html)" on _MathWorld_, or entry [A078437](http://oeis.org/A078437) in the Sloane's OEIS.2012-11-21
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    Here are some first $8250$ digits. http://dl.dropbox.com/u/5805435/Alternating_prime_sum.txt Unfortunately, it is not $\dfrac{\gamma}2$2012-11-21
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    The Inverse Symbolic Calculator http://isc.carma.newcastle.edu.au/index has no suggestions.2012-11-21

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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.