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A078437 Decimal expansion of sum of alternating series of reciprocals of primes. 13
2, 6, 9, 6, 0, 6, 3, 5, 1, 9, 7, 1, 6, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Verified and extended by Chris K. Caldwell and Jud McCranie.

Next two terms are most likely 4 and 5. - Robert Price, Sep 13 2011

From Jon E. Schoenfield, Nov 25 2018: (Start)

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()).

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:

                         h(k) =

           k  (f(k-2) + 2*f(k-1) + f(k))/4

  ==========  ============================

           2     0.29166666666666666...

           4     0.28095238095238095...

           8     0.26875529011751921...

          16     0.27058892362329746...

          32     0.27009944617052797...

          64     0.26963971020080367...

         128     0.26959147218377685...

         256     0.26959653902072193...

         512     0.26960402179695026...

        1024     0.26960568606633210...

        2048     0.26960649673621509...

        4096     0.26960645080540929...

        8192     0.26960627432070023...

       16384     0.26960633643086948...

       32768     0.26960634835658329...

       65536     0.26960635083481533...

      131072     0.26960635144743392...

      262144     0.26960635199009778...

      524288     0.26960635199971603...

     1048576     0.26960635195886861...

     2097152     0.26960635197214933...

     4194304     0.26960635197019215...

     8388608     0.26960635197186919...

    16777216     0.26960635197171149...

    33554432     0.26960635197146884...

    67108864     0.26960635197167534...

   134217728     0.26960635197167145...

   268435456     0.26960635197166927...

   536870912     0.26960635197167200...

  1073741824     0.26960635197167416...

  2147483648     0.26960635197167454...

  4294967296     0.26960635197167462... (End)

REFERENCES

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.

LINKS

Table of n, a(n) for n=0..13.

Eric Weisstein's World of Mathematics, Prime Sums

Eric Weisstein's World of Mathematics, Prime Zeta Function

Wikipedia, Prime Zeta Function

EXAMPLE

1/2 - 1/3 + 1/5 - 1/7 + 1/11 - 1/13 + ... = 0.26960635197167...

MATHEMATICA

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-Fran├žois Alcover, Sep 02 2015 *)

CROSSREFS

Cf. A242301, A242302, A242303, A242304.

Sequence in context: A021375 A190407 A057052 * A235997 A155678 A134946

Adjacent sequences:  A078434 A078435 A078436 * A078438 A078439 A078440

KEYWORD

cons,hard,more,nonn

AUTHOR

G. L. Honaker, Jr., Dec 31 2002

EXTENSIONS

Values of a(11)-a(14) = 7,1,6,7 from Robert Price, Sep 13 2011

STATUS

approved

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Last modified October 23 09:28 EDT 2019. Contains 328345 sequences. (Running on oeis4.)