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Decimal expansion of Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).
3

%I #11 Oct 14 2017 05:51:07

%S 7,0,5,8,4,0,7,1,3,1,1,3,0,4,2,1,7,9,5,3,6,1,3,3,3,5,9,6,6,1,2,9,7,6,

%T 1,3,3,0,4,7,6,7,1,0,8,7,8,0,4,8,5,9,6,5,5,3,2,3,0,5,9,8,9,7,6,4,3,1,

%U 9,0,9,5,2,2,1,5,1,1,3,7,5,0,2,3,9,0,8,5,3,6,6,5,0,2,5,5,8,4,3,2,7,4,7,5,1,0,6,2,0,5,2,4,3,3,0,0,3,0,7,8

%N Decimal expansion of Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

%C This constant plus A293383 equals log(3), due to the identity:

%C Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 2/3, y = 1/2.

%F Constant: Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

%F Constant: log(3) - Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

%e Constant t = 0.7058407131130421795361333596612976133047671087804859655323059...

%e such that

%e t = 3/(1*2*(3-1)) - 3^2/(2*4*(3*2-1)^2) + 3^3/(3*8*(3*2^2-1)^3) - 3^4/(4*16*(3*2^3-1)^4) + 3^5/(5*32*(3*2^4-1)^5) - 3^6/(6*64*(3*2^5-1)^6) + 3^7/(7*128*(3*2^6-1)^7) - 3^8/(8*256*(3*2^7-1)^8) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...

%e More explicitly,

%e t = 3/(1*2*2) - 9/(2*4*5^2) + 27/(3*8*11^3) - 81/(4*16*23^4) + 243/(5*32*47^5) - 729/(6*64*95^6) + 2187/(7*128*191^7) - 6561/(8*256*383^8) + 19683/(9*512*767^9) - 59049/(10*1024*1535^10) + 177147/(11*2048*3071^11) - 531441/(12*4096*6143^12) +...

%e Also,

%e log(3) - t = (2^2 - 3)/(1*3*2) + (2^3 - 3)^2/(2*3^2*2^4) + (2^4 - 3)^3/(3*3^3*2^9) + (2^5 - 3)^4/(4*3^4*2^16) + (2^6 - 3)^5/(5*3^5*2^25) + (2^7 - 3)^6/(6*3^6*2^36) + (2^8 - 3)^7/(7*3^7*2^49) + (2^9 - 3)^8/(8*3^8*2^64) + (2^10 - 3)^9/(9*3^9*2^81) +...+ (2^(n+1) - 3)^n/(n * 3^n * 2^(n^2)) +...

%o (PARI) {t = suminf(n=1, -1.*(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n) )}

%o for(n=1,120, print1(floor(10^n*t)%10,", "))

%Y Cf. A293383, A293381, A293382, A292178, A292179, A002391 (log 3).

%K nonn,cons

%O 0,1

%A _Paul D. Hanna_, Oct 13 2017