24.2

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
24.2.E1	${\displaystyle{\displaystyle\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!}}}$ \frac{t}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullinumberB{n}\frac{t^{n}}{n!}	${\displaystyle{\displaystyle|t|<2\pi}}$	(t)/(exp(t)- 1) = sum(bernoulli(n)*((t)^(n))/(factorial(n)), n = 0..infinity)	Divide[t,Exp[t]- 1] == Sum[BernoulliB[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 6]	Successful [Tested: 6]
24.2#Ex1	${\displaystyle{\displaystyle B_{2n+1}=0}}$ \BernoullinumberB{2n+1} = 0		bernoulli(2*n + 1) = 0	BernoulliB[2*n + 1] == 0	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
24.2#Ex2	${\displaystyle{\displaystyle(-1)^{n+1}B_{2n}>0}}$ (-1)^{n+1}\BernoullinumberB{2n} > 0		(- 1)^(n + 1)* bernoulli(2*n) > 0	(- 1)^(n + 1)* BernoulliB[2*n] > 0	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 3]
24.2.E3	${\displaystyle{\displaystyle\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\left(x\right)\frac{t^{n}}{n!}}}$ \frac{te^{xt}}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullipolyB{n}@{x}\frac{t^{n}}{n!}	${\displaystyle{\displaystyle|t|<2\pi}}$	(t*exp(x*t))/(exp(t)- 1) = sum(bernoulli(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity)	Divide[t*Exp[x*t],Exp[t]- 1] == Sum[BernoulliB[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 18]	Successful [Tested: 18]
24.2.E4	${\displaystyle{\displaystyle B_{n}=B_{n}\left(0\right)}}$ \BernoullinumberB{n} = \BernoullipolyB{n}@{0}		bernoulli(n) = bernoulli(n, 0)	BernoulliB[n] == BernoulliB[n, 0]	Successful	Successful	-	Successful [Tested: 3]
24.2.E5	${\displaystyle{\displaystyle B_{n}\left(x\right)=\sum_{k=0}^{n}{n\choose k}B_{k}x^{n-k}}}$ \BernoullipolyB{n}@{x} = \sum_{k=0}^{n}{n\choose k}\BernoullinumberB{k}x^{n-k}		bernoulli(n, x) = sum(binomial(n,k)*bernoulli(k)*(x)^(n - k), k = 0..n)	BernoulliB[n, x] == Sum[Binomial[n,k]*BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 9]	Successful [Tested: 9]
24.2.E6	${\displaystyle{\displaystyle\frac{2e^{t}}{e^{2t}+1}=\sum_{n=0}^{\infty}E_{n}\frac{t^{n}}{n!}}}$ \frac{2e^{t}}{e^{2t}+1} = \sum_{n=0}^{\infty}\EulernumberE{n}\frac{t^{n}}{n!}	${\displaystyle{\displaystyle|t|<\tfrac{1}{2}\pi}}$	(2*exp(t))/(exp(2*t)+ 1) = sum(euler(n)*((t)^(n))/(factorial(n)), n = 0..infinity)	Divide[2*Exp[t],Exp[2*t]+ 1] == Sum[EulerE[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]	Missing Macro Error	Successful	-	Successful [Tested: 4]
24.2#Ex3	${\displaystyle{\displaystyle E_{2n+1}=0}}$ \EulernumberE{2n+1} = 0		euler(2*n + 1) = 0	EulerE[2*n + 1] == 0	Missing Macro Error	Failure	-	Successful [Tested: 3]
24.2#Ex4	${\displaystyle{\displaystyle(-1)^{n}E_{2n}>0}}$ (-1)^{n}\EulernumberE{2n} > 0		(- 1)^(n)* euler(2*n) > 0	(- 1)^(n)* EulerE[2*n] > 0	Missing Macro Error	Failure	-	Successful [Tested: 3]
24.2.E8	${\displaystyle{\displaystyle\frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!}}}$ \frac{2e^{xt}}{e^{t}+1} = \sum_{n=0}^{\infty}\EulerpolyE{n}@{x}\frac{t^{n}}{n!}	${\displaystyle{\displaystyle|t|<\pi}}$	(2*exp(x*t))/(exp(t)+ 1) = sum(euler(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity)	Divide[2*Exp[x*t],Exp[t]+ 1] == Sum[EulerE[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 18]
24.2.E9	${\displaystyle{\displaystyle E_{n}=2^{n}E_{n}\left(\tfrac{1}{2}\right)}}$ \EulernumberE{n} = 2^{n}\EulerpolyE{n}@{\tfrac{1}{2}}		euler(n) = (2)^(n)* euler(n, (1)/(2))	EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]	Missing Macro Error	Successful	-	Successful [Tested: 3]
24.2.E10	${\displaystyle{\displaystyle E_{n}\left(x\right)=\sum_{k=0}^{n}{n\choose k}\frac{E_{k}}{2^{k}}(x-\tfrac{1}{2})^{n-k}}}$ \EulerpolyE{n}@{x} = \sum_{k=0}^{n}{n\choose k}\frac{\EulernumberE{k}}{2^{k}}(x-\tfrac{1}{2})^{n-k}		euler(n, x) = sum(binomial(n,k)*(euler(k))/((2)^(k))*(x -(1)/(2))^(n - k), k = 0..n)	EulerE[n, x] == Sum[Binomial[n,k]*Divide[EulerE[k],(2)^(k)]*(x -Divide[1,2])^(n - k), {k, 0, n}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [3 / 9] Result: Indeterminate Test Values: {Rule[n, 1], Rule[x, 0.5]} Result: Indeterminate Test Values: {Rule[n, 2], Rule[x, 0.5]} ... skip entries to safe data