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%I #29 Jan 16 2022 22:10:10
%S 3,7,4,4,8,5,1,8,7,9,7,4,7,4,6,1,6,3,2,1,7,0,9,4,0,8,6
%N Decimal expansion of Sum_{k>=1} (-1)^(k+1)/(k*prime(k)).
%C Alternating sum of the reciprocals of the products of k and the k-th prime.
%C From _Jon E. Schoenfield_, Jan 15 2021: (Start)
%C The second Mathematica program appears to compute partial sums through k=10^10 and includes a comment that it is "good for the first 27 digits". For the (10^10-1)st, (10^10)th and (10^10+1)st partial sums, I get
%C .
%C k prime(k) S(k) = k-th partial sum
%C =========== ============ ====================================
%C 9999999999 252097800611 0.3744851879747461632172924014592...
%C 10000000000 252097800623 0.3744851879747461632168957300096...
%C 10000000001 252097800629 0.3744851879747461632172924014591...
%C .
%C Since the sum is alternating, successive partial sums oscillate around a central curve, and taking the mean of the k-th and (k+1)st partial sums gives an estimate of the value around which the sum appears to be converging. Those means for k = 9999999999 and 10000000000 are
%C .
%C k (S(k) + S(k+1))/2
%C =========== =====================================
%C 9999999999 0.37448518797474616321709406573443...
%C 10000000000 0.37448518797474616321709406573440...
%C .
%C Of course, even though these results happen to agree through their first 31 significant digits, they are certainly nowhere near sufficient to establish the first 31 significant digits of the infinite sum (since the sequence of mean partial sums tends to meander). However, these results are clearly lower than the value given in the Data (i.e., 0.374485187974746163217094086), so (if correct) they would appear to indicate either that the last two terms in the Data are incorrect or that those terms were obtained using partial sums significantly beyond the (10^10)th. (End)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%F Sum_{k>=1} (-1)^(k+1)/(k*prime(k)) = 1/(1*2) - 1/(2*3) + 1/(3*5) - 1/(4*7) + 1/(5*11) - ... .
%e 0.374485187974746163217094086...
%t RealDigits[N[-Sum[(-1)^k/(k*Prime[k]), {k, 1, 8*10^6}], 30]][[1]] (* _G. C. Greubel_, Nov 22 2016 *)
%t s = 0; k = 1; p = 2; While[k =< 10^10, s = N[s - (-1)^k/(k*p), 48]; k++; p = NextPrime@p]; RealDigits[s, 10, 20] (* good for the first 27 digits *) (* _Robert G. Wilson v_, Mar 07 2019 *)
%Y See the following for alternating sums of reciprocals of primes, composites, and related expressions: A078437 (primes), A242301 (primes^2), A242302 (primes^3), A242303 (primes^4), A242304 (primes^5), A269229 (composites), A275110 (composites excluding prime powers), A275712 (nonprimes), A276494 (composites^2).
%K nonn,cons,hard,more
%O 0,1
%A _Jon E. Schoenfield_, Nov 20 2016
%E Edited and a(21)-a(26) from _Robert G. Wilson v_, Mar 07 2019