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%I #29 Feb 26 2020 06:44:05
%S 0,4,3,9,0,8,9,6,3,3,5,1,3,0,8
%N Decimal expansion of the sum of the alternating series of reciprocals of squares of composite numbers.
%C From _Jon E. Schoenfield_, Feb 02 2018: (Start)
%C Let S(j) be the partial sum through the j-th term of the alternating series, i.e., S(j) = -Sum_{m=1..j} (-1)^m/A002808(m)^2. The sequence of real values S(2*i-1) for i >= 1, i.e., of partial sums 1/16, 1/16 - 1/36 + 1/64, 1/16 - 1/36 + 1/64 - 1/81 + 1/100, ... (each of which ends with a positive term) will approach the limit from above, while the sequence of real values S(2*i) for i >= 1, i.e., of partial sums 1/16 - 1/36, 1/16 - 1/36 + 1/64 - 1/81, 1/16 - 1/36 + 1/64 - 1/81 + 1/100 - 1/144, ... (each of which ends with a negative term) will approach the limit from below. Let S'(j) = (S(j-1) + S(j))/2; equivalently, S'(j) = -(Sum_{m=1..j-1} (-1)^m/A002808(m)^2 + (1/2)*(-1)^j/A002808(j)^2), so S'(j) can be viewed as an adjusted version of S(j), adjusted by using only half of the final term of S(j). At large values of j, successive values of S'(j) will fluctuate very little compared to the differences between successive values of S(j), because the averaging of successive values of S(j), which are above the limit at each odd value of j and below the limit by very nearly the same amount at each even value of j, causes the values of S'(j) to trace a path midway between that traced by the S(j) values for odd j and those for even j.
%C Moreover, similar to the situation at A275712, it can be verified that the values of S'(j) themselves fall into three sharply distinct real-valued subsequences: one that converges toward the limit from above and consists of those values where both j and the j-th composite number (i.e., the square root of the reciprocal of the last term in S(j)) are even; one that converges toward the limit from below and consists of those values where j is odd and the j-th composite number is even; and one that stays very near the middle, converging even more rapidly toward the limit, and consisting of all those values where the j-th composite number is odd (regardless of the parity of j). The values in this last subsequence converge very rapidly; see the table in the Example section, which lists values of S'(c_k) where c_k is the smallest odd composite number > 2^k. (End)
%F Sum_{k>=1} (-1)^(k+1)/A062312(k+1) = 1/16 - 1/36 + 1/64 - 1/81...
%F Sum_{k>=1} (-1)^(k+1)/((A002808(k))^2) = 1/(4^2) - 1/(6^2) + 1/(8^2) - 1/(9^2)...
%e Equals 0.04390896...
%e From _Jon E. Schoenfield_, Feb 02 2018: (Start)
%e Let S'(j) = -(Sum_{m=1..j-1} (-1)^m/A002808(m)^2 + (1/2)*(-1)^j/A002808(j)^2) and let c_k be the smallest odd composite > 2^k; S'(c_k) quickly converges to a limit, as illustrated below:
%e k c_k S'(c_k)
%e == ======== ===============================
%e 3 9 0.04417438271604938271604938...
%e 4 21 0.04390073853615520282186948...
%e 5 33 0.04390758368090798391978693...
%e 6 65 0.04390888269964319809070094...
%e 7 129 0.04390902395888932501501797...
%e 8 259 0.04390896620540588616012725...
%e 9 513 0.04390896281303069589885533...
%e 10 1025 0.04390896330786777379414334...
%e 11 2049 0.04390896335161701542401577...
%e 12 4097 0.04390896335102793828470954...
%e 13 8193 0.04390896335127457473079624...
%e 14 16385 0.04390896335131185998890588...
%e 15 32769 0.04390896335130880417881285...
%e 16 65541 0.04390896335130852088789156...
%e 17 131073 0.04390896335130852182995244...
%e 18 262145 0.04390896335130852702777625...
%e 19 524289 0.04390896335130852688659318...
%e 20 1048577 0.04390896335130852691520992...
%e 21 2097153 0.04390896335130852691785136...
%e 22 4194305 0.04390896335130852691786707...
%e 23 8388609 0.04390896335130852691787563...
%e 24 16777217 0.04390896335130852691787421...
%e 25 33554433 0.04390896335130852691787435...
%e ...
%e Extending this several steps farther, it becomes apparent that the limit is 0.04390896335130852691787434869606... (End)
%o (Sage)
%o compositessq = (i for i in NN if i>3 and not i.is_prime())
%o s = RLF(0); s
%o RealField(110)(s)
%o for i in range(0, 50000000): s += (-1)^i / next(compositessq)^2
%o print(s)
%Y Cf. A002162 (of natural numbers), A072691 (of natural numbers squared), A078437 (of primes), A242301 (of primes squared), A269229 (of composite numbers), A275110 (of composite numbers with distinct prime factors), A275712 (of nonprime numbers).
%Y Cf. A275712.
%K nonn,more,cons
%O 1,2
%A _Terry D. Grant_, Sep 05 2016
%E a(9)-a(15) from _Robert Price_, Nov 14 2016
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