24.9

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
24.9.E1	${\displaystyle{\displaystyle|B_{2n}|>|B_{2n}\left(x\right)|}}$ |\BernoullinumberB{2n}| > |\BernoullipolyB{2n}@{x}|	${\displaystyle{\displaystyle 1>x,x>0}}$	abs(bernoulli(2*n)) > abs(bernoulli(2*n, x))	Abs[BernoulliB[2*n]] > Abs[BernoulliB[2*n, x]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
24.9.E2	${\displaystyle{\displaystyle(2-2^{1-2n})|B_{2n}|\geq|B_{2n}\left(x\right)-B_{2n}|}}$ (2-2^{1-2n})|\BernoullinumberB{2n}| \geq |\BernoullipolyB{2n}@{x}-\BernoullinumberB{2n}|	${\displaystyle{\displaystyle 1\geq x,x\geq 0}}$	(2 - (2)^(1 - 2*n))*abs(bernoulli(2*n)) >= abs(bernoulli(2*n, x)- bernoulli(2*n))	(2 - (2)^(1 - 2*n))*Abs[BernoulliB[2*n]] >= Abs[BernoulliB[2*n, x]- BernoulliB[2*n]]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
24.9.E3	${\displaystyle{\displaystyle 4^{-n}|E_{2n}|>(-1)^{n}E_{2n}\left(x\right)}}$ 4^{-n}|\EulernumberE{2n}| > (-1)^{n}\EulerpolyE{2n}@{x}		(4)^(- n)*abs(euler(2*n)) > (- 1)^(n)* euler(2*n, x)	(4)^(- n)*Abs[EulerE[2*n]] > (- 1)^(n)* EulerE[2*n, x]	Missing Macro Error	Failure	Skip - symbolical successful subtest	Failed [4 / 9] Result: False Test Values: {Rule[n, 1], Rule[x, 0.5]} Result: False Test Values: {Rule[n, 2], Rule[x, 0.5]} ... skip entries to safe data
24.9.E3	${\displaystyle{\displaystyle(-1)^{n}E_{2n}\left(x\right)>0}}$ (-1)^{n}\EulerpolyE{2n}@{x} > 0		(- 1)^(n)* euler(2*n, x) > 0	(- 1)^(n)* EulerE[2*n, x] > 0	Failure	Failure	Failed [5 / 9] Result: 0. < -.7500000000 Test Values: {x = 3/2, n = 1} Result: 0. < -.1875000000 Test Values: {x = 3/2, n = 2} ... skip entries to safe data	Failed [5 / 9] Result: False Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: False Test Values: {Rule[n, 2], Rule[x, 1.5]} ... skip entries to safe data
24.9.E4	${\displaystyle{\displaystyle\frac{2(2n+1)!}{(2\pi)^{2n+1}}>(-1)^{n+1}B_{2n+1}\left(x\right)}}$ \frac{2(2n+1)!}{(2\pi)^{2n+1}} > (-1)^{n+1}\BernoullipolyB{2n+1}@{x}		(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1)) > (- 1)^(n + 1)* bernoulli(2*n + 1, x)	Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)] > (- 1)^(n + 1)* BernoulliB[2*n + 1, x]	Failure	Failure	Successful [Tested: 3]	Failed [4 / 9] Result: False Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: False Test Values: {Rule[n, 3], Rule[x, 1.5]} ... skip entries to safe data
24.9.E4	${\displaystyle{\displaystyle(-1)^{n+1}B_{2n+1}\left(x\right)>0}}$ (-1)^{n+1}\BernoullipolyB{2n+1}@{x} > 0		(- 1)^(n + 1)* bernoulli(2*n + 1, x) > 0	(- 1)^(n + 1)* BernoulliB[2*n + 1, x] > 0	Failure	Failure	Failed [3 / 3] Result: 0. < -.3125000000 Test Values: {x = 3/2, n = 2} Result: 0. < 0. Test Values: {x = 1/2, n = 2} ... skip entries to safe data	Failed [5 / 9] Result: False Test Values: {Rule[n, 2], Rule[x, 1.5]} Result: False Test Values: {Rule[n, 1], Rule[x, 0.5]} ... skip entries to safe data
24.9.E5	${\displaystyle{\displaystyle\frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2}>(-1)^{n}E_{2n-1}\left(x\right)}}$ \frac{4(2n-1)!}{\pi^{2n}}\frac{2^{2n}-1}{2^{2n}-2} > (-1)^{n}\EulerpolyE{2n-1}@{x}		(4*factorial(2*n - 1))/((Pi)^(2*n))*((2)^(2*n)- 1)/((2)^(2*n)- 2) > (- 1)^(n)* euler(2*n - 1, x)	Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Divide[(2)^(2*n)- 1,(2)^(2*n)- 2] > (- 1)^(n)* EulerE[2*n - 1, x]	Failure	Failure	Failed [1 / 9] Result: 2.250000000 < .2639824007 Test Values: {x = 2, n = 2}	Failed [1 / 9] Result: False Test Values: {Rule[n, 2], Rule[x, 2]}
24.9.E5	${\displaystyle{\displaystyle(-1)^{n}E_{2n-1}\left(x\right)>0}}$ (-1)^{n}\EulerpolyE{2n-1}@{x} > 0		(- 1)^(n)* euler(2*n - 1, x) > 0	(- 1)^(n)* EulerE[2*n - 1, x] > 0	Failure	Failure	Failed [7 / 9] Result: 0. < -1. Test Values: {x = 3/2, n = 1} Result: 0. < -.6250000000e-1 Test Values: {x = 3/2, n = 3} ... skip entries to safe data	Failed [7 / 9] Result: False Test Values: {Rule[n, 1], Rule[x, 1.5]} Result: False Test Values: {Rule[n, 3], Rule[x, 1.5]} ... skip entries to safe data
24.9.E6	${\displaystyle{\displaystyle 5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}>(-1)^{n+1}B_{2n}}}$ 5\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n} > (-1)^{n+1}\BernoullinumberB{2n}		5*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n) > (- 1)^(n + 1)* bernoulli(2*n)	5*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n) > (- 1)^(n + 1)* BernoulliB[2*n]	Failure	Failure	Failed [1 / 3] Result: .1666666667 < .1215223702 Test Values: {n = 1}	Failed [1 / 3] Result: False Test Values: {Rule[n, 1]}
24.9.E6	${\displaystyle{\displaystyle(-1)^{n+1}B_{2n}>4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}}}$ (-1)^{n+1}\BernoullinumberB{2n} > 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n}		(- 1)^(n + 1)* bernoulli(2*n) > 4*sqrt(Pi*n)*((n)/(Pi*exp(1)))^(2*n)	(- 1)^(n + 1)* BernoulliB[2*n] > 4*Sqrt[Pi*n]*(Divide[n,Pi*E])^(2*n)	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
24.9.E7	${\displaystyle{\displaystyle 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}\right)>(-1)^{n}E_{2n}}}$ 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}\right) > (-1)^{n}\EulernumberE{2n}		8*sqrt((n)/(Pi))*((4*n)/(Pi*exp(1)))^(2*n)*(1 +(1)/(12*n)) > (- 1)^(n)* euler(2*n)	8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)*(1 +Divide[1,12*n]) > (- 1)^(n)* EulerE[2*n]	Missing Macro Error	Failure	-	Successful [Tested: 3]
24.9.E7	${\displaystyle{\displaystyle(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}}}$ (-1)^{n}\EulernumberE{2n} > 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}		(- 1)^(n)* euler(2*n) > 8*sqrt((n)/(Pi))*((4*n)/(Pi*exp(1)))^(2*n)	(- 1)^(n)* EulerE[2*n] > 8*Sqrt[Divide[n,Pi]]*(Divide[4*n,Pi*E])^(2*n)	Missing Macro Error	Failure	-	Successful [Tested: 3]
24.9.E8	${\displaystyle{\displaystyle\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}}\geq(-1)^{n+1}B_{2n}\geq\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}}}$ \frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{\beta-2n}} \geq (-1)^{n+1}\BernoullinumberB{2n}\geq\frac{2(2n)!}{(2\pi)^{2n}}\frac{1}{1-2^{-2n}}		(2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(beta - 2*n)) >= (- 1)^(n + 1)* bernoulli(2*n) >= (2*factorial(2*n))/((2*Pi)^(2*n))*(1)/(1 - (2)^(- 2*n))	Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(\[Beta]- 2*n)] >= (- 1)^(n + 1)* BernoulliB[2*n] >= Divide[2*(2*n)!,(2*Pi)^(2*n)]*Divide[1,1 - (2)^(- 2*n)]	Failure	Failure	Error	Failed [2 / 9] Result: False Test Values: {Rule[n, 1], Rule[β, 0.5]} Result: GreaterEqual[DirectedInfinity[], 0.16666666666666666] Test Values: {Rule[n, 1], Rule[β, 2]}
24.9.E9	${\displaystyle{\displaystyle\beta=2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}}}$ \beta = 2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}}		beta = 2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2))	\[Beta] == 2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]]	Aborted	Failure	Failed [3 / 3] Result: .850806174 Test Values: {beta = 3/2} Result: -.1491938260 Test Values: {beta = 1/2} ... skip entries to safe data	Failed [3 / 3] Result: 0.850806175200028 Test Values: {Rule[β, 1.5]} Result: -0.149193824799972 Test Values: {Rule[β, 0.5]} ... skip entries to safe data
24.9.E9	${\displaystyle{\displaystyle 2+\frac{\ln\left(1-6\pi^{-2}\right)}{\ln 2}=0.6491\dots}}$ 2+\frac{\ln@{1-6\pi^{-2}}}{\ln@@{2}} = 0.6491\dots		2 +(ln(1 - 6*(Pi)^(- 2)))/(ln(2)) = 0.6491	2 +Divide[Log[1 - 6*(Pi)^(- 2)],Log[2]] == 0.6491	Error	Failure	Skip - symbolical successful subtest	Successful [Tested: 1]
24.9.E10	${\displaystyle{\displaystyle\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}}}$ \frac{4^{n+1}(2n)!}{\pi^{2n+1}} > (-1)^{n}\EulernumberE{2n}		((4)^(n + 1)*factorial(2*n))/((Pi)^(2*n + 1)) > (- 1)^(n)* euler(2*n)	Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)] > (- 1)^(n)* EulerE[2*n]	Missing Macro Error	Failure	-	Successful [Tested: 3]
24.9.E10	${\displaystyle{\displaystyle(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}\frac{1}{1+3^{-1-2n}}}}$ (-1)^{n}\EulernumberE{2n} > \frac{4^{n+1}(2n)!}{\pi^{2n+1}}\frac{1}{1+3^{-1-2n}}		(- 1)^(n)* euler(2*n) > ((4)^(n + 1)*factorial(2*n))/((Pi)^(2*n + 1))*(1)/(1 + (3)^(- 1 - 2*n))	(- 1)^(n)* EulerE[2*n] > Divide[(4)^(n + 1)*(2*n)!,(Pi)^(2*n + 1)]*Divide[1,1 + (3)^(- 1 - 2*n)]	Missing Macro Error	Failure	-	Successful [Tested: 3]