%I #59 Feb 25 2024 11:10:05
%S 2,6,9,6,0,6,3,5,1,9,7,1,6,7
%N Decimal expansion of sum of alternating series of reciprocals of primes.
%C Verified and extended by _Chris K. Caldwell_ and _Jud McCranie_.
%C Next two terms are most likely 4 and 5. - _Robert Price_, Sep 13 2011
%C From _Jon E. Schoenfield_, Nov 25 2018: (Start)
%C Let f(k) be the k-th partial sum of the alternating series, i.e., f(k) = Sum_{j=1..k} ((-1)^(j+1))/prime(j). At large values of k, successive first differences f(k) - f(k-1) = ((-1)^(k+1))/prime(k) are alternatingly positive and negative and are nearly the same in absolute value, so f(k) is alternatingly above (for odd k) or below (for even k) the value of the much smoother function g(k) = (f(k-1) + f(k))/2 (a two-point moving average of the function f()).
%C Additionally, since the first differences f(k) - f(k-1) are decreasing in absolute value, g(k) will be less than both g(k-1) and g(k+1) for odd k, and greater than both for even k; i.e., g(), although much smoother than f(), is also alternatingly below or above the value of the still smoother function h(k) = (g(k-1) + g(k))/2 = ((f(k-2) + f(k-1))/2 + (f(k-1) + f(k))/2)/2 = (f(k-2) + 2*f(k-1) + f(k))/4. Evaluated at k = 2^m for m = 1, 2, 3, ..., the values of h(k) converge fairly quickly toward the limit of the alternating series:
%C h(k) =
%C k (f(k-2) + 2*f(k-1) + f(k))/4
%C ========== ============================
%C 2 0.29166666666666666...
%C 4 0.28095238095238095...
%C 8 0.26875529011751921...
%C 16 0.27058892362329746...
%C 32 0.27009944617052797...
%C 64 0.26963971020080367...
%C 128 0.26959147218377685...
%C 256 0.26959653902072193...
%C 512 0.26960402179695026...
%C 1024 0.26960568606633210...
%C 2048 0.26960649673621509...
%C 4096 0.26960645080540929...
%C 8192 0.26960627432070023...
%C 16384 0.26960633643086948...
%C 32768 0.26960634835658329...
%C 65536 0.26960635083481533...
%C 131072 0.26960635144743392...
%C 262144 0.26960635199009778...
%C 524288 0.26960635199971603...
%C 1048576 0.26960635195886861...
%C 2097152 0.26960635197214933...
%C 4194304 0.26960635197019215...
%C 8388608 0.26960635197186919...
%C 16777216 0.26960635197171149...
%C 33554432 0.26960635197146884...
%C 67108864 0.26960635197167534...
%C 134217728 0.26960635197167145...
%C 268435456 0.26960635197166927...
%C 536870912 0.26960635197167200...
%C 1073741824 0.26960635197167416...
%C 2147483648 0.26960635197167454...
%C 4294967296 0.26960635197167462... (End)
%C The above mentioned average functions can also be written g(k) = f(k) + (-1)^k/prime(k)/2 and h(k) = g(k) + (-1)^k (1/prime(k) - 1/prime(k-1))/4 = f(k) + (-1)^k (3/prime(k) - 1/prime(k-1))/4. - _M. F. Hasler_, Feb 20 2024
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_zeta_function">Prime Zeta Function</a>
%F c = lim_{n -> oo} A024530(n)/A002110(n). - _M. F. Hasler_, Feb 20 2024
%e 1/2 - 1/3 + 1/5 - 1/7 + 1/11 - 1/13 + ... = 0.26960635197167...
%t s = NSum[ p=Prime[k//Round]; (-1)^k/p, {k, 1, Infinity}, WorkingPrecision -> 30, NSumTerms -> 5*10^7, Method -> "AlternatingSigns"]; RealDigits[s, 10, 14] // First (* _Jean-François Alcover_, Sep 02 2015 *)
%o (PARI) L=2^N=1; h=List([1/4, 1/6 + S=.5-1/o=3]); forprime(p=o+1,oo, S+=(-1)^L/p; L--|| print([L=2^N++, p, S, listput(h, S+(3/p-1/o)/4)]); o=p) \\ in PARI version > 2.13 listput() may not return the element so one must add +h[#h]
%o A(x,n=#x)=(x[n]*x[n-2]-x[n-1]^2)/(x[n]+x[n-2]-2*x[n-1]) \\ This is Aitken's Delta-square extrapolation for the last 3 elements of the list x. One can check that the extrapolation is useful for the sequence of raw partial sums (f(2^k)), but not for the smooth/average sequence (h(2^k)). - _M. F. Hasler_, Feb 20 2024
%Y Cf. A242301, A242302, A242303, A242304.
%Y Cf. A024530 (numerator of partial sums), A002110 (denominators: primorials).
%K cons,hard,more,nonn
%O 0,1
%A _G. L. Honaker, Jr._, Dec 31 2002
%E Values of a(11)-a(14) = 7,1,6,7 from _Robert Price_, Sep 13 2011