Bernoulli and Euler Polynomials - 24.8 Series Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
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Mathematica
24.8.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n}@{x} = (-1)^{n+1}\frac{2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos@{2\pi kx}}{k^{2n}}}
\BernoullipolyB{2n}@{x} = (-1)^{n+1}\frac{2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos@{2\pi kx}}{k^{2n}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(2*n, x) = (- 1)^(n + 1)*(2*factorial(2*n))/((2*Pi)^(2*n))*sum((cos(2*Pi*k*x))/((k)^(2*n)), k = 1..infinity)

BernoulliB[2*n, x] == (- 1)^(n + 1)*Divide[2*(2*n)!,(2*Pi)^(2*n)]*Sum[Divide[Cos[2*Pi*k*x],(k)^(2*n)], {k, 1, Infinity}, GenerateConditions->None]
Aborted Failure
Failed [6 / 9]
Result: 1.000000000
Test Values: {x = 3/2, n = 1}


Result: .5000000000
Test Values: {x = 3/2, n = 2}

... skip entries to safe data
Failed [6 / 9]
Result: Complex[1.0, 0.0]
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Complex[0.5000000000000001, 0.0]
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.8.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{2n+1}@{x} = (-1)^{n+1}\frac{2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac{\sin@{2\pi kx}}{k^{2n+1}}}
\BernoullipolyB{2n+1}@{x} = (-1)^{n+1}\frac{2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac{\sin@{2\pi kx}}{k^{2n+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(2*n + 1, x) = (- 1)^(n + 1)*(2*factorial(2*n + 1))/((2*Pi)^(2*n + 1))*sum((sin(2*Pi*k*x))/((k)^(2*n + 1)), k = 1..infinity)

BernoulliB[2*n + 1, x] == (- 1)^(n + 1)*Divide[2*(2*n + 1)!,(2*Pi)^(2*n + 1)]*Sum[Divide[Sin[2*Pi*k*x],(k)^(2*n + 1)], {k, 1, Infinity}, GenerateConditions->None]
Aborted Failure -
Failed [6 / 9]
Result: Complex[0.75, 0.0]
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Complex[0.3125, 0.0]
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.8.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n}@{x} = (-1)^{n}\frac{4(2n)!}{\pi^{2n+1}}\sum_{k=0}^{\infty}\frac{\sin@{(2k+1)\pi x}}{(2k+1)^{2n+1}}}
\EulerpolyE{2n}@{x} = (-1)^{n}\frac{4(2n)!}{\pi^{2n+1}}\sum_{k=0}^{\infty}\frac{\sin@{(2k+1)\pi x}}{(2k+1)^{2n+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
euler(2*n, x) = (- 1)^(n)*(4*factorial(2*n))/((Pi)^(2*n + 1))*sum((sin((2*k + 1)*Pi*x))/((2*k + 1)^(2*n + 1)), k = 0..infinity)

EulerE[2*n, x] == (- 1)^(n)*Divide[4*(2*n)!,(Pi)^(2*n + 1)]*Sum[Divide[Sin[(2*k + 1)*Pi*x],(2*k + 1)^(2*n + 1)], {k, 0, Infinity}, GenerateConditions->None]
Aborted Failure
Failed [6 / 9]
Result: .5000000000+0.*I
Test Values: {x = 3/2, n = 1}


Result: .1249999998+0.*I
Test Values: {x = 3/2, n = 2}

... skip entries to safe data
Failed [6 / 9]
Result: Complex[0.4999999999999999, -6.717074394942855*^-17]
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Complex[0.1250000000000001, -1.3482715791848248*^-17]
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.8.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{2n-1}@{x} = (-1)^{n}\frac{4(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}\frac{\cos@{(2k+1)\pi x}}{(2k+1)^{2n}}}
\EulerpolyE{2n-1}@{x} = (-1)^{n}\frac{4(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}\frac{\cos@{(2k+1)\pi x}}{(2k+1)^{2n}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
euler(2*n - 1, x) = (- 1)^(n)*(4*factorial(2*n - 1))/((Pi)^(2*n))*sum((cos((2*k + 1)*Pi*x))/((2*k + 1)^(2*n)), k = 0..infinity)

EulerE[2*n - 1, x] == (- 1)^(n)*Divide[4*(2*n - 1)!,(Pi)^(2*n)]*Sum[Divide[Cos[(2*k + 1)*Pi*x],(2*k + 1)^(2*n)], {k, 0, Infinity}, GenerateConditions->None]
Aborted Failure
Failed [6 / 9]
Result: 1.
Test Values: {x = 3/2, n = 1}


Result: .2500000000
Test Values: {x = 3/2, n = 2}

... skip entries to safe data
Failed [6 / 9]
Result: Complex[1.0, -2.3810929344395102*^-33]
Test Values: {Rule[n, 1], Rule[x, 1.5]}


Result: Complex[0.25000000000000006, 5.146963577016199*^-33]
Test Values: {Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.8.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{4n+2} = (8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}{e^{2\pi k}-1}}
\BernoullinumberB{4n+2} = (8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}{e^{2\pi k}-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(4*n + 2) = (8*n + 4)*sum(((k)^(4*n + 1))/(exp(2*Pi*k)- 1), k = 1..infinity)

BernoulliB[4*n + 2] == (8*n + 4)*Sum[Divide[(k)^(4*n + 1),Exp[2*Pi*k]- 1], {k, 1, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 1]
Failed [3 / 3]
Result: Plus[0.023809523809523808, Times[-12.0, NSum[Times[Power[Plus[-1, Power[E, Times[2, k, Pi]]], -1], Power[k, 5]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1]}


Result: Plus[0.07575757575757576, Times[-20.0, NSum[Times[Power[Plus[-1, Power[E, Times[2, k, Pi]]], -1], Power[k, 9]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2]}

... skip entries to safe data
24.8.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullinumberB{2n} = \frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{\pi k}+(-1)^{k+n}}}
\BernoullinumberB{2n} = \frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{\pi k}+(-1)^{k+n}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(2*n) = ((- 1)^(n + 1)* 4*n)/((2)^(2*n)- 1)*sum(((k)^(2*n - 1))/(exp(Pi*k)+(- 1)^(k + n)), k = 1..infinity)

BernoulliB[2*n] == Divide[(- 1)^(n + 1)* 4*n,(2)^(2*n)- 1]*Sum[Divide[(k)^(2*n - 1),Exp[Pi*k]+(- 1)^(k + n)], {k, 1, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 1]
Failed [3 / 3]
Result: Plus[0.16666666666666666, Times[-1.3333333333333333, NSum[Times[Power[Plus[Power[-1, Plus[1, k]], Power[E, Times[k, Pi]]], -1], k]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 1]}


Result: Plus[-0.03333333333333333, Times[0.5333333333333333, NSum[Times[Power[Plus[Power[-1, Plus[2, k]], Power[E, Times[k, Pi]]], -1], Power[k, 3]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[n, 2]}

... skip entries to safe data
24.8.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\BernoullinumberB{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}\right) = \alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1}}
\frac{\BernoullinumberB{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}\right) = \alpha^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)^{n}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{2\beta k}-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(bernoulli(2*n))/(4*n)*((alpha)^(n)-(- beta)^(n)) = (alpha)^(n)* sum(((k)^(2*n - 1))/(exp(2*alpha*k)- 1), k = 1..infinity)-(- beta)^(n)* sum(((k)^(2*n - 1))/(exp(2*beta*k)- 1), k = 1..infinity)

Divide[BernoulliB[2*n],4*n]*(\[Alpha]^(n)-(- \[Beta])^(n)) == \[Alpha]^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Alpha]*k]- 1], {k, 1, Infinity}, GenerateConditions->None]-(- \[Beta])^(n)* Sum[Divide[(k)^(2*n - 1),Exp[2*\[Beta]*k]- 1], {k, 1, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [6 / 9]
Result: 1.443014212
Test Values: {alpha = 3/2, beta = 1/2, n = 2}


Result: -.7774267192e-1
Test Values: {alpha = 3/2, beta = 2, n = 2}

... skip entries to safe data
 Skipped - Because timed out
24.8.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {}\EulernumberE{2n} = (-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh@{\tfrac{1}{2}\pi k}}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1}}
{}\EulernumberE{2n} = (-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh@{\tfrac{1}{2}\pi k}}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
*euler(2*n) = (- 1)^(n)* sum(((k)^(2*n))/(cosh((1)/(2)*Pi*k)), k = 1..infinity)- 4*sum(((- 1)^(k)*(2*k + 1)^(2*n))/(exp(2*Pi*(2*k + 1))- 1), k = 0..infinity)

*EulerE[2*n] == (- 1)^(n)* Sum[Divide[(k)^(2*n),Cosh[Divide[1,2]*Pi*k]], {k, 1, Infinity}, GenerateConditions->None]- 4*Sum[Divide[(- 1)^(k)*(2*k + 1)^(2*n),Exp[2*Pi*(2*k + 1)]- 1], {k, 0, Infinity}, GenerateConditions->None]
Translation Error Translation Error - -
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