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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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forums.xkcd.com/… –  alex.jordan Nov 21 '12 at 3:33
    
More directly, see "Prime sums" on MathWorld, or entry A078437 in the Sloane's OEIS. –  Sasha Nov 21 '12 at 3:35
    
Here are some first $8250$ digits. dl.dropbox.com/u/5805435/Alternating_prime_sum.txt Unfortunately, it is not $\dfrac{\gamma}2$ –  user17762 Nov 21 '12 at 3:43
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The Inverse Symbolic Calculator isc.carma.newcastle.edu.au/index has no suggestions. –  Gerry Myerson Nov 21 '12 at 4:48

1 Answer 1

up vote 1 down vote accepted

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.

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