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Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).
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%I #25 Sep 28 2017 22:26:02

%S 4,4,6,6,8,4,3,0,7

%N Decimal expansion of sum of alternating series of reciprocals of Ramanujan primes, Sum_{n>=1} (1/R_n)(-1)^(n-1), where R_n is the n-th Ramanujan prime, A104272(n).

%C Computed 0.446684 for n = 1 to 65536, using Open Office Calc. Next digit expected to be between 2 and 3.

%C By computing all Ramanujan primes less than 10^9, we find that about 9 decimal places of the sum should be correct: 0.446684307 (truncated, not rounded). The following table shows the number of Ramanujan primes between powers of 10 and the sum of the alternating reciprocals of those primes.

%C 1 1 0.50000000000000000

%C 2 9 -0.05765566386047510

%C 3 62 0.00388002010130731

%C 4 487 0.00050881775862179

%C 5 3900 -0.00004384563815649

%C 6 32501 -0.00000552572415587

%C 7 279106 0.00000045427780897

%C 8 2444255 0.00000005495474474

%C 9 21731345 -0.00000000549864067

%C Total: 0.44668430669928564 - _T. D. Noe_, May 08 2011

%C Let E_n denote the error after the first n terms in the series. Then by the Alternating Series Test, 1/R_{n+1} - 1/R_{n+2} < E_n < 1/R_{n+1}. [_Jonathan Sondow_, May 10 2011]

%H J. Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, arXiv:0907.5232 [math.NT], 2009-2010.

%H J. Sondow, <a href="http://www.jstor.org/stable/40391170">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly 116 (2009), 630-635.

%H J. Sondow, J. W. Nicholson, and T. D. Noe, <a href="http://arxiv.org/abs/1105.2249"> Ramanujan Primes: Bounds, Runs, Twins, and Gaps</a>, J. Integer Seq. 14 (2011) Article 11.6.2

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>

%F Sum_{n>=1} (-1)^(n-1)(1/R_n), where R_n is the n-th Ramanujan prime, A104272(n).

%e 0.446684307...

%Y Cf. A104272, A085548, A078437, A190124.

%K nonn,cons,more

%O 0,1

%A _John W. Nicholson_, May 07 2011

%E Definition corrected by _Jonathan Sondow_, May 10 2011