25.8

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.8.E1	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1\right)=1}}$ \sum_{k=2}^{\infty}\left(\Riemannzeta@{k}-1\right) = 1		sum(Zeta(k)- 1, k = 2..infinity) = 1	Sum[Zeta[k]- 1, {k, 2, Infinity}, GenerateConditions->None] == 1	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
25.8.E2	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(k+1)!}\left(\zeta\left(s+k\right)-1\right)=\Gamma\left(s-1\right)}}$ \sum_{k=0}^{\infty}\frac{\EulerGamma@{s+k}}{(k+1)!}\left(\Riemannzeta@{s+k}-1\right) = \EulerGamma@{s-1}	${\displaystyle{\displaystyle\Re(s+k)>0,\Re(s-1)>0}}$	sum((GAMMA(s + k))/(factorial(k + 1))*(Zeta(s + k)- 1), k = 0..infinity) = GAMMA(s - 1)	Sum[Divide[Gamma[s + k],(k + 1)!]*(Zeta[s + k]- 1), {k, 0, Infinity}, GenerateConditions->None] == Gamma[s - 1]	Failure	Aborted	Successful [Tested: 1]	Skipped - Because timed out
25.8.E3	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta\left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta\left(s\right)}}$ \sum_{k=0}^{\infty}\frac{\Pochhammersym{s}{k}\Riemannzeta@{s+k}}{k!2^{s+k}} = (1-2^{-s})\Riemannzeta@{s}	${\displaystyle{\displaystyle s\neq 1}}$	sum((pochhammer(s, k)*Zeta(s + k))/(factorial(k)*(2)^(s + k)), k = 0..infinity) = (1 - (2)^(- s))*Zeta(s)	Sum[Divide[Pochhammer[s, k]*Zeta[s + k],(k)!*(2)^(s + k)], {k, 0, Infinity}, GenerateConditions->None] == (1 - (2)^(- s))*Zeta[s]	Failure	Successful	Failed [1 / 6] Result: -.1666666667 Test Values: {s = -2}	Successful [Tested: 6]
25.8.E4	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right)}}$ \sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\Riemannzeta@{nk}-1) = \ln@{\prod_{j=0}^{n-1}\EulerGamma@{2-e^{(2j+1)\pi i/n}}}		sum(((- 1)^(k))/(k)*(Zeta(n*k)- 1), k = 1..infinity) = ln(product(GAMMA(2 - exp((2*j + 1)*Pi*I/n)), j = 0..n - 1))	Sum[Divide[(- 1)^(k),k]*(Zeta[n*k]- 1), {k, 1, Infinity}, GenerateConditions->None] == Log[Product[Gamma[2 - Exp[(2*j + 1)*Pi*I/n]], {j, 0, n - 1}, GenerateConditions->None]]	Failure	Failure	Successful [Tested: 1]	Failed [1 / 3] Result: Plus[-0.6931471805599453, NSum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[k]]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[n, 1]}
25.8.E5	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}=-\gamma z-z\psi\left(1-z\right)}}$ \sum_{k=2}^{\infty}\Riemannzeta@{k}z^{k} = -\EulerConstant z-z\digamma@{1-z}	${\displaystyle{\displaystyle|z|<1}}$	sum(Zeta(k)*(z)^(k), k = 2..infinity) = - gamma*z - z*Psi(1 - z)	Sum[Zeta[k]*(z)^(k), {k, 2, Infinity}, GenerateConditions->None] == - EulerGamma*z - z*PolyGamma[1 - z]	Failure	Successful	Successful [Tested: 1]	Successful [Tested: 1]
25.8.E6	${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\zeta\left(2k\right)z^{2k}=-\tfrac{1}{2}\pi z\cot\left(\pi z\right)}}$ \sum_{k=0}^{\infty}\Riemannzeta@{2k}z^{2k} = -\tfrac{1}{2}\pi z\cot@{\pi z}	${\displaystyle{\displaystyle|z|<1}}$	sum(Zeta(2*k)*(z)^(2*k), k = 0..infinity) = -(1)/(2)*Pi*z*cot(Pi*z)	Sum[Zeta[2*k]*(z)^(2*k), {k, 0, Infinity}, GenerateConditions->None] == -Divide[1,2]*Pi*z*Cot[Pi*z]	Failure	Failure	Error	Failed [1 / 1] Result: Plus[4.8091767343044744*^-17, NSum[Times[Power[0.5, Times[2, k]], Zeta[Times[2, k]]] Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, 0.5]}
25.8.E7	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^{k}=-\gamma z+\ln\Gamma\left(1-z\right)}}$ \sum_{k=2}^{\infty}\frac{\Riemannzeta@{k}}{k}z^{k} = -\EulerConstant z+\ln@@{\EulerGamma@{1-z}}	${\displaystyle{\displaystyle|z|<1,\Re(1-z)>0}}$	sum((Zeta(k))/(k)*(z)^(k), k = 2..infinity) = - gamma*z + ln(GAMMA(1 - z))	Sum[Divide[Zeta[k],k]*(z)^(k), {k, 2, Infinity}, GenerateConditions->None] == - EulerGamma*z + Log[Gamma[1 - z]]	Failure	Successful	Successful [Tested: 1]	Successful [Tested: 1]
25.8.E8	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right)}}$ \sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{k}z^{2k} = \ln@{\frac{\pi z}{\sin@{\pi z}}}	${\displaystyle{\displaystyle|z|<1}}$	sum((Zeta(2*k))/(k)*(z)^(2*k), k = 1..infinity) = ln((Pi*z)/(sin(Pi*z)))	Sum[Divide[Zeta[2*k],k]*(z)^(2*k), {k, 1, Infinity}, GenerateConditions->None] == Log[Divide[Pi*z,Sin[Pi*z]]]	Failure	Successful	Successful [Tested: 1]	Successful [Tested: 1]
25.8.E9	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2}}$ \sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)2^{2k}} = \frac{1}{2}-\frac{1}{2}\ln@@{2}		sum((Zeta(2*k))/((2*k + 1)*(2)^(2*k)), k = 1..infinity) = (1)/(2)-(1)/(2)*ln(2)	Sum[Divide[Zeta[2*k],(2*k + 1)*(2)^(2*k)], {k, 1, Infinity}, GenerateConditions->None] == Divide[1,2]-Divide[1,2]*Log[2]	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
25.8.E10	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4\pi^{2}}\zeta\left(3\right)}}$ \sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)(2k+2)2^{2k}} = \frac{1}{4}-\frac{7}{4\pi^{2}}\Riemannzeta@{3}		sum((Zeta(2*k))/((2*k + 1)*(2*k + 2)*(2)^(2*k)), k = 1..infinity) = (1)/(4)-(7)/(4*(Pi)^(2))*Zeta(3)	Sum[Divide[Zeta[2*k],(2*k + 1)*(2*k + 2)*(2)^(2*k)], {k, 1, Infinity}, GenerateConditions->None] == Divide[1,4]-Divide[7,4*(Pi)^(2)]*Zeta[3]	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]