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A276524
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Decimal expansion of the apparent (see Comments) limit of the sum of the alternating series 1/prime(1) - 2/prime(2) + 3/prime(3) - 4/prime(4) + ...
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1
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OFFSET
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-1,1
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COMMENTS
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I.e., -Sum_{k>=1} (-1)^k*k/prime(k). The Prime Sums page at Mathworld (see Links) says that it is not known whether Sum_{k>=1} (-1)^k*k/prime(k) (which is the negative of the sum treated here) converges; it cites the References entries for Guy, Erdős, and Finch.
However, if we define the j-th partial sum S(j) = -Sum_{k=1..j} (-1)^k*k/prime(k) and the interpolated partial sum S'(j) = (S(j-1) + S(j))/2 = -Sum_{k=1..j-1} (-1)^k*k/prime(k) - (1/2)(-1)^j*j/prime(j) (which is the same as S(j) except that S'(j) uses only half of the j-th term) and observe the behavior of S'(j) as j increases, it seems clear that the sum converges to 0.052160..., or possibly 0.052161... (see the charts under Links).
If it were the case that this alternating series actually did diverge, there does not seem to be anything about it that would bias the sum either in the positive or negative direction; since the numerator is simply k (the term number) and the denominator is simply the k-th prime, it would seem that the sum, if it were going to diverge, would do so by increasing in amplitude as successive terms alternately yielded partial sums above and below some approximate centerline (as opposed to the kind of divergence seen in, e.g., a nonalternating sum such as Sum_{k>=1} 1/prime(k)). Also, a plot of the curve along which the partial sums S(j) lie for even values of j, the curve along which they lie for odd values of j, and the interpolated sum S'(j) = (S(j-1) + S(j))/2 does not seem at all consistent with a widening-amplitude type of divergence; instead, the amplitude is clearly narrowing, and is doing so in a way that seems nearly perfectly symmetrical across the S'(j) vs. j curve. Note also that the variability of S'(j) vs. j over small intervals of j (i.e., its "roughness") is decreasing as j increases. All these observations seem consistent with an alternating sum that is actually converging, albeit slowly, toward a limiting value halfway between the odd-j and even-j curves traced by plots of S(j) vs. j.
Note that this sequence, if it does indeed converge, does so extremely slowly; observe how far apart the odd-j and even-j curves are at larger values of prime(j), yet how close each curve's slope is to zero. At the right end of the chart, the last point on the even-j curve is for j = 37607912018, and the corresponding prime is 999999999989 (the last prime < 10^12), so S(j-1) - S(j) there is 37607912018/999999999989 = 0.037607912... To get the odd-j and even-j curves to agree even to a single significant digit (presumably, 0.05...) would require an enormous number of terms:
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j S(j-1) - S(j)
----- -------------
10^10 0.039667...
10^20 0.020441...
10^30 0.013822...
10^40 0.010454...
10^50 0.008410...
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REFERENCES
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Finch, S. R. "Meissel-Mertens Constants." Sec. 2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 94-98, 2003.
Guy, R. K. "A Series and a Sequence Involving Primes." Sec. E7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 203, 1994.
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LINKS
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Eric Weisstein's World of Mathematics, Prime Sums (see eq. 8)
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FORMULA
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-Sum_{k>=1} (-1)^k * k/prime(k).
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EXAMPLE
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0.052160...
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MATHEMATICA
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(* This simple script gives only 3 presumably correct digits: 0.052158... *) -NSum[(-1)^k*k/Prime[Round[k]], {k, 1, Infinity}, Method -> "AlternatingSigns", NSumTerms -> 10^8, WorkingPrecision -> 100] // Chop (* Jean-François Alcover, Dec 05 2016 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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