The question if there is an upper bound known for Brun's constant was discussed briefly here: http://gowers.wordpress.com/2009/05/22/what-is-wolfram-alpha-good-for/ but no sure answer was given.

So I thought I'd ask the question here. Can one get any upper bound for the sum of the reciprocals of the twin primes?

isprecisely the sum of the reciprocals of the twin primes. – Pete L. Clark May 1 '10 at 21:51