Thank you both for your time.
Mr. Laurie, your chapter in the SIAM 100-digit challenge was one of the first places I initially checked for a solution. I'm new to PARI and could not find a working implementation of Levin transforms in any language, so thank you very much for the script. It gets 16 places, which is enough for the OEIS to identify it (
https://oeis.org/search?q=0.0840695087276559). Is there a systematic way to pick out where to stop the summation for similar problems? I assume we picked 40 here because the values obtained didn't agree with the expected value past there, but what if one did not know that beforehand?
Mr. Cohen, the second solution works perfectly for me, but for some reason I cannot initialize the Monien summation. I get:
> tab=sumnummonieninit(1/2)
*** at top-level: tab=sumnummonieninit(1/2
*** ^--------------------
*** sumnummonieninit: domain error in sumnummonieninit: a+b <= -1
*** Break loop: type 'break' to go back to GP prompt
break>
The proof I was shown is a bit tricky but essentially uses only basic complex analysis. I could post it when I finish writing it up.
Kevin.