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%I #16 Jun 25 2021 23:15:16
%S 0,1,1,3,-4,-5,3,13,5,2,-10,-3,4,2,-42,-12,3,8,-9,-2,6,-50,5,-67,-5,7,
%T 12,-401,-2,-2,3,3,-4,-6,3,3,-12,-3,-2,2,2,-5,-6
%N Continued fraction for Euler-Mascheroni constant with convergents 0/1, 1/1, 1/2, 4/7, etc., which lie between the monotonically increasing series given by (Sum_{k=1..n} 1/k - Sum_{k=n..n^2} 1/k) and the monotonically decreasing series (Sum_{k=1..n} 1/k - Sum_{k=n..n^2-1} 1/k), both of which converge to gamma. Thus each p/q in the sequence lies within 1/q^2 of gamma.
%C Series derived from def. gamma = lim(Sum_{k=1..n} 1/k - log(n)) by noting that 2*gamma = 2*Sum_{k=1..n} 1/k - 2*log(n) (ignoring limit) and also gamma = Sum_{k=1..n^2} 1/k - log(n^2), then gamma = 2*gamma - gamma gets rid of the log term and the series consists of all rational terms. The decreasing series was found by accident. The proofs for both are straightforward. The PARI program uses the first term of the Euler-Maclaurin summation and gamma itself for the upper and lower bounds.
%o (PARI) pconv=vector(43); qconv=vector(43); cf=vector(43); fract=vector(43); pconv[1]=0; pconv[2]=1; pconv[3]=1; pconv[4]=4; qconv[1]=1; qconv[2]=1; qconv[3]=2; qconv[4]=7; cf[1]=0; cf[2]=1; cf[3]=1; cf[4]=3; fract[1]=0/1; fract[2]=1/1; fract[3]=1/2; fract[4]=4/7; for(k=5,43, tst=0; cfm=1; until(tst==1, pp = cfm * pconv[k - 1] + pconv[k - 2]; pn = cfm * pconv[k - 1] - pconv[k - 2]; qp = cfm * qconv[k - 1] + qconv[k - 2]; qn = cfm * qconv[k - 1] - qconv[k - 2]; slp = pp/qp; sln = pn/qn; if(((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) ||
%o (slp - Euler < 1/(3 * qp^2) && slp - Euler > 0)) || ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0)), pconv[k] = ((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) || (slp - Euler < 1/(3 * qp^2) && slp - Euler > 0))*pp + ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0))*pn; qconv[k] = ((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) ||
%o (slp - Euler < 1/(3 * qp^2) && slp - Euler > 0))*qp + ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0))*qn; fract[k] = pconv[k]/qconv[k]; cf[k] = ((Euler - slp < 2/(3 * qp^2) && Euler - slp > 0) || (slp - Euler < 1/(3 * qp^2) && slp - Euler > 0))*cfm - ((Euler - sln < 2/(3 * qn^2) && Euler - sln > 0) || (sln - Euler < 1/(3 * qn^2) && sln - Euler > 0))*cfm; tst = 1, cfm = cfm + 1)); write("eulwritefile.txt","Convergents: ",fract); write("eulwritefile.txt","continued fraction: ",cf); write("eulwritefile.txt","sln: ",sln); write("eulwritefile.txt","slp: ",slp))
%Y Cf. A002852.
%K sign
%O 0,4
%A Joseph G. Johnson (jjohnson1253(AT)hotmail.com), Jun 12 2010
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