Asymptotic Approximations - 2.10 Sums and Sequences

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.10.E3 S ⁢ ( n ) = ∑ j = 1 n j ⁢ ln ⁡ j 𝑆 𝑛 superscript subscript 𝑗 1 𝑛 𝑗 𝑗 {\displaystyle{\displaystyle S(n)=\sum_{j=1}^{n}j\ln j}}
S(n) = \sum_{j=1}^{n}j\ln@@{j}		 

S(n) = sum(j*ln(j), j = 1..n)

S[n] == Sum[j*Log[j], {j, 1, n}, GenerateConditions->None]
Failure Failure
Failed [30 / 30]
Result: .8660254040+.5000000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .345756447+1.*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -2.084055016+1.500000000*I
Test Values: {S = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -.5000000000+.8660254040*I
Test Values: {S = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.34575644644898684, 0.9999999999999999]
Test Values: {Rule[n, 2], Rule[S, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
2.10.E4 S ⁢ ( n ) = 1 2 ⁢ n 2 ⁢ ln ⁡ n - 1 4 ⁢ n 2 + 1 2 ⁢ n ⁢ ln ⁡ n + 1 12 ⁢ ln ⁡ n + C + ∑ s = 2 m - 1 ( - B 2 ⁢ s ) 2 ⁢ s ⁢ ( 2 ⁢ s - 1 ) ⁢ ( 2 ⁢ s - 2 ) ⁢ 1 n 2 ⁢ s - 2 + R m ⁢ ( n ) 𝑆 𝑛 1 2 superscript 𝑛 2 𝑛 1 4 superscript 𝑛 2 1 2 𝑛 𝑛 1 12 𝑛 𝐶 superscript subscript 𝑠 2 𝑚 1 Bernoulli-number-B 2 𝑠 2 𝑠 2 𝑠 1 2 𝑠 2 1 superscript 𝑛 2 𝑠 2 subscript 𝑅 𝑚 𝑛 {\displaystyle{\displaystyle S(n)=\tfrac{1}{2}n^{2}\ln n-\tfrac{1}{4}n^{2}+% \tfrac{1}{2}n\ln n+\tfrac{1}{12}\ln n+C+\sum_{s=2}^{m-1}\frac{(-B_{2s})}{2s(2s% -1)(2s-2)}\frac{1}{n^{2s-2}}+R_{m}(n)}}
S(n) = \tfrac{1}{2}n^{2}\ln@@{n}-\tfrac{1}{4}n^{2}+\tfrac{1}{2}n\ln@@{n}+\tfrac{1}{12}\ln@@{n}+C+\sum_{s=2}^{m-1}\frac{(-\BernoullinumberB{2s})}{2s(2s-1)(2s-2)}\frac{1}{n^{2s-2}}+R_{m}(n)		 

S(n) = (1)/(2)*(n)^(2)* ln(n)-(1)/(4)*(n)^(2)+(1)/(2)*n*ln(n)+(1)/(12)*ln(n)+ C + sum((- bernoulli(2*s))/(2*s*(2*s - 1)*(2*s - 2))*(1)/((n)^(2*s - 2)), s = 2..m - 1)+ R[m](n)

S[n] == Divide[1,2]*(n)^(2)* Log[n]-Divide[1,4]*(n)^(2)+Divide[1,2]*n*Log[n]+Divide[1,12]*Log[n]+ C + Sum[Divide[- BernoulliB[2*s],2*s*(2*s - 1)*(2*s - 2)]*Divide[1,(n)^(2*s - 2)], {s, 2, m - 1}, GenerateConditions->None]+ Subscript[R, m][n]
Failure Aborted Error Skipped - Because timed out
2.10.E6 C = γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 𝐶 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 {\displaystyle{\displaystyle C=\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{% \zeta'\left(2\right)}{2\pi^{2}}}}
C = \frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}}		 

C = (gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2))

C == Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)]
Failure Failure
Failed [10 / 10]
Result: .6172709270+.5000000000*I
Test Values: {C = 1/2*3^(1/2)+1/2*I}


Result: -.7487544770+.8660254040*I
Test Values: {C = -1/2+1/2*I*3^(1/2)}


Result: .2512455230-.8660254040*I
Test Values: {C = 1/2-1/2*I*3^(1/2)}


Result: -1.114779881-.5000000000*I
Test Values: {C = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[0.6172709267506544, 0.49999999999999994]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.748754477033784, 0.8660254037844387]
Test Values: {Rule[C, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
2.10.E6 γ + ln ⁡ ( 2 ⁢ π ) 12 - ζ ′ ⁡ ( 2 ) 2 ⁢ π 2 = 1 12 - ζ ′ ⁡ ( - 1 ) 2 𝜋 12 diffop Riemann-zeta 1 2 2 superscript 𝜋 2 1 12 diffop Riemann-zeta 1 1 {\displaystyle{\displaystyle\frac{\gamma+\ln\left(2\pi\right)}{12}-\frac{\zeta% '\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'\left(-1\right)}}
\frac{\EulerConstant+\ln@{2\pi}}{12}-\frac{\Riemannzeta'@{2}}{2\pi^{2}} = \frac{1}{12}-\Riemannzeta'@{-1}		 

(gamma + ln(2*Pi))/(12)-(subs( temp=2, diff( Zeta(temp), temp$(1) ) ))/(2*(Pi)^(2)) = (1)/(12)- subs( temp=- 1, diff( Zeta(temp), temp$(1) ) )

Divide[EulerGamma + Log[2*Pi],12]-Divide[D[Zeta[temp], {temp, 1}]/.temp-> 2,2*(Pi)^(2)] == Divide[1,12]- (D[Zeta[temp], {temp, 1}]/.temp-> - 1)
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
2.10.E9 ∑ j = 1 n - 1 u j ⁢ v j = U n - 1 ⁢ v n + ∑ j = 1 n - 1 U j ⁢ ( v j - v j + 1 ) superscript subscript 𝑗 1 𝑛 1 subscript 𝑢 𝑗 subscript 𝑣 𝑗 subscript 𝑈 𝑛 1 subscript 𝑣 𝑛 superscript subscript 𝑗 1 𝑛 1 subscript 𝑈 𝑗 subscript 𝑣 𝑗 subscript 𝑣 𝑗 1 {\displaystyle{\displaystyle\sum_{j=1}^{n-1}u_{j}v_{j}=U_{n-1}v_{n}+\sum_{j=1}% ^{n-1}U_{j}(v_{j}-v_{j+1})}}
\sum_{j=1}^{n-1}u_{j}v_{j} = U_{n-1}v_{n}+\sum_{j=1}^{n-1}U_{j}(v_{j}-v_{j+1})		 

sum(u[j]*v[j], j = 1..n - 1) = U[n - 1]*v[n]+ sum(U[j]*(v[j]- v[j + 1]), j = 1..n - 1)
 
Sum[Subscript[u, j]*Subscript[v, j], {j, 1, n - 1}, GenerateConditions->None] == Subscript[U, n - 1]*Subscript[v, n]+ Sum[Subscript[U, j]*(Subscript[v, j]- Subscript[v, j + 1]), {j, 1, n - 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
2.10.E11 S ⁢ ( α , β , n ) = ∑ j = 1 n - 1 e i ⁢ j ⁢ β ⁢ j α 𝑆 𝛼 𝛽 𝑛 superscript subscript 𝑗 1 𝑛 1 superscript 𝑒 𝑖 𝑗 𝛽 superscript 𝑗 𝛼 {\displaystyle{\displaystyle S(\alpha,\beta,n)=\sum_{j=1}^{n-1}e^{ij\beta}j^{% \alpha}}}
S(\alpha,\beta,n) = \sum_{j=1}^{n-1}e^{ij\beta}j^{\alpha}		 

S(alpha , beta , n) = sum(exp(I*j*beta)*(j)^(alpha), j = 1..n - 1)
 
S[\[Alpha], \[Beta], n] == Sum[Exp[I*j*\[Beta]]*(j)^\[Alpha], {j, 1, n - 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
2.10.E12 | S ⁢ ( α , β , n ) | ≤ ∑ j = 1 n - 1 j α 𝑆 𝛼 𝛽 𝑛 superscript subscript 𝑗 1 𝑛 1 superscript 𝑗 𝛼 {\displaystyle{\displaystyle|S(\alpha,\beta,n)|\leq\sum_{j=1}^{n-1}j^{\alpha}}}
|S(\alpha,\beta,n)| \leq \sum_{j=1}^{n-1}j^{\alpha}		 

abs(S(alpha , beta , n)) <= sum((j)^(alpha), j = 1..n - 1)

Abs[S[\[Alpha], \[Beta], n]] <= Sum[(j)^\[Alpha], {j, 1, n - 1}, GenerateConditions->None]
Error Failure - Error
2.10.E13 U j = e i ⁢ β ⁢ ( e i ⁢ j ⁢ β - 1 ) / ( e i ⁢ β - 1 ) subscript 𝑈 𝑗 superscript 𝑒 𝑖 𝛽 superscript 𝑒 𝑖 𝑗 𝛽 1 superscript 𝑒 𝑖 𝛽 1 {\displaystyle{\displaystyle U_{j}=e^{i\beta}(e^{ij\beta}-1)/(e^{i\beta}-1)}}
U_{j} = e^{i\beta}(e^{ij\beta}-1)/(e^{i\beta}-1)		 

U[j] = exp(I*beta)*(exp(I*j*beta)- 1)/(exp(I*beta)- 1)
 
Subscript[U, j] == Exp[I*\[Beta]]*(Exp[I*j*\[Beta]]- 1)/(Exp[I*\[Beta]]- 1)
 Skipped - no semantic math Skipped - no semantic math - -
2.10.E14 S ⁢ ( α , β , n ) = e i ⁢ β e i ⁢ β - 1 ⁢ ( e i ⁢ ( n - 1 ) ⁢ β ⁢ n α - 1 + ∑ j = 1 n - 1 e i ⁢ j ⁢ β ⁢ ( j α - ( j + 1 ) α ) ) 𝑆 𝛼 𝛽 𝑛 superscript 𝑒 𝑖 𝛽 superscript 𝑒 𝑖 𝛽 1 superscript 𝑒 𝑖 𝑛 1 𝛽 superscript 𝑛 𝛼 1 superscript subscript 𝑗 1 𝑛 1 superscript 𝑒 𝑖 𝑗 𝛽 superscript 𝑗 𝛼 superscript 𝑗 1 𝛼 {\displaystyle{\displaystyle S(\alpha,\beta,n)=\frac{e^{i\beta}}{e^{i\beta}-1}% \left(e^{i(n-1)\beta}n^{\alpha}-1+\sum_{j=1}^{n-1}e^{ij\beta}\left(j^{\alpha}-% (j+1)^{\alpha}\right)\right)}}
S(\alpha,\beta,n) = \frac{e^{i\beta}}{e^{i\beta}-1}\left(e^{i(n-1)\beta}n^{\alpha}-1+\sum_{j=1}^{n-1}e^{ij\beta}\left(j^{\alpha}-(j+1)^{\alpha}\right)\right)		 

S(alpha , beta , n) = (exp(I*beta))/(exp(I*beta)- 1)*(exp(I*(n - 1)*beta)*(n)^(alpha)- 1 + sum(exp(I*j*beta)*((j)^(alpha)-(j + 1)^(alpha)), j = 1..n - 1))
 
S[\[Alpha], \[Beta], n] == Divide[Exp[I*\[Beta]],Exp[I*\[Beta]]- 1]*(Exp[I*(n - 1)*\[Beta]]*(n)^\[Alpha]- 1 + Sum[Exp[I*j*\[Beta]]*((j)^\[Alpha]-(j + 1)^\[Alpha]), {j, 1, n - 1}, GenerateConditions->None])
 Skipped - no semantic math Skipped - no semantic math - -
2.10.E19 F 2 0 ⁡ ( - ; 1 , 1 ; x ) = ∑ j = 0 ∞ x j ( j ! ) 3 Gauss-hypergeometric-pFq 0 2 1 1 𝑥 superscript subscript 𝑗 0 superscript 𝑥 𝑗 superscript 𝑗 3 {\displaystyle{\displaystyle{{}_{0}F_{2}}\left(-;1,1;x\right)=\sum_{j=0}^{% \infty}\frac{x^{j}}{(j!)^{3}}}}
\genhyperF{0}{2}@{-}{1,1}{x} = \sum_{j=0}^{\infty}\frac{x^{j}}{(j!)^{3}}		 

hypergeom([-], [1 , 1], x) = sum(((x)^(j))/((factorial(j))^(3)), j = 0..infinity)

HypergeometricPFQ[{-}, {1 , 1}, x] == Sum[Divide[(x)^(j),((j)!)^(3)], {j, 0, Infinity}, GenerateConditions->None]
Error Failure - Error
2.10.E21 cot ⁡ ( π ⁢ t ) 2 ⁢ i = - 1 2 - 1 e - 2 ⁢ π ⁢ i ⁢ t - 1 𝜋 𝑡 2 𝑖 1 2 1 superscript 𝑒 2 𝜋 𝑖 𝑡 1 {\displaystyle{\displaystyle\frac{\cot\left(\pi t\right)}{2i}=-\frac{1}{2}-% \frac{1}{e^{-2\pi it}-1}}}
\frac{\cot@{\pi t}}{2i} = -\frac{1}{2}-\frac{1}{e^{-2\pi it}-1}		 

(cot(Pi*t))/(2*I) = -(1)/(2)-(1)/(exp(- 2*Pi*I*t)- 1)

Divide[Cot[Pi*t],2*I] == -Divide[1,2]-Divide[1,Exp[- 2*Pi*I*t]- 1]
Successful Successful -
Failed [2 / 6]
Result: Indeterminate
Test Values: {Rule[t, -2]}


Result: Indeterminate
Test Values: {Rule[t, 2]}

2.10.E21 - 1 2 - 1 e - 2 ⁢ π ⁢ i ⁢ t - 1 = 1 2 + 1 e 2 ⁢ π ⁢ i ⁢ t - 1 1 2 1 superscript 𝑒 2 𝜋 𝑖 𝑡 1 1 2 1 superscript 𝑒 2 𝜋 𝑖 𝑡 1 {\displaystyle{\displaystyle-\frac{1}{2}-\frac{1}{e^{-2\pi it}-1}=\frac{1}{2}+% \frac{1}{e^{2\pi it}-1}}}
-\frac{1}{2}-\frac{1}{e^{-2\pi it}-1} = \frac{1}{2}+\frac{1}{e^{2\pi it}-1}		 

-(1)/(2)-(1)/(exp(- 2*Pi*I*t)- 1) = (1)/(2)+(1)/(exp(2*Pi*I*t)- 1)

-Divide[1,2]-Divide[1,Exp[- 2*Pi*I*t]- 1] == Divide[1,2]+Divide[1,Exp[2*Pi*I*t]- 1]
Successful Successful -
Failed [2 / 6]
Result: Indeterminate
Test Values: {Rule[t, -2]}


Result: Indeterminate
Test Values: {Rule[t, 2]}

2.10.E23 F 2 0 ⁡ ( - ; 1 , 1 ; x ) = ∫ - 1 / 2 ∞ x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t + 2 ⁢ ℜ ⁡ ∫ - 1 / 2 i ⁢ ∞ x t ( Γ ⁡ ( t + 1 ) ) 3 ⁢ d t e - 2 ⁢ π ⁢ i ⁢ t - 1 Gauss-hypergeometric-pFq 0 2 1 1 𝑥 superscript subscript 1 2 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 2 superscript subscript 1 2 𝑖 superscript 𝑥 𝑡 superscript Euler-Gamma 𝑡 1 3 𝑡 superscript 𝑒 2 𝜋 𝑖 𝑡 1 {\displaystyle{\displaystyle{{}_{0}F_{2}}\left(-;1,1;x\right)=\int_{-1/2}^{% \infty}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\mathrm{d}t+2\Re\int_{-1/2}^{% i\infty}\frac{x^{t}}{(\Gamma\left(t+1\right))^{3}}\frac{\mathrm{d}t}{e^{-2\pi it% }-1}}}
\genhyperF{0}{2}@{-}{1,1}{x} = \int_{-1/2}^{\infty}\frac{x^{t}}{(\EulerGamma@{t+1})^{3}}\diff{t}+2\realpart@@{\int_{-1/2}^{i\infty}\frac{x^{t}}{(\EulerGamma@{t+1})^{3}}\frac{\diff{t}}{e^{-2\pi it}-1}}		 ℜ ⁡ ( t + 1 ) > 0 𝑡 1 0 {\displaystyle{\displaystyle\Re(t+1)>0}} 
hypergeom([-], [1 , 1], x) = int(((x)^(t))/((GAMMA(t + 1))^(3)), t = - 1/2..infinity)+ 2*Re(int(((x)^(t))/((GAMMA(t + 1))^(3))*(1)/(exp(- 2*Pi*I*t)- 1), t = - 1/2..I*infinity))

HypergeometricPFQ[{-}, {1 , 1}, x] == Integrate[Divide[(x)^(t),(Gamma[t + 1])^(3)], {t, - 1/2, Infinity}, GenerateConditions->None]+ 2*Re[Integrate[Divide[(x)^(t),(Gamma[t + 1])^(3)]*Divide[1,Exp[- 2*Pi*I*t]- 1], {t, - 1/2, I*Infinity}, GenerateConditions->None]]
Error Failure - Error
2.10.E35 g n = ( 2 π ⁢ sin ⁡ α ) 1 / 2 ⁢ Γ ⁡ ( n + 1 2 ) n ! ⁢ cos ⁡ ( n ⁢ α + 1 2 ⁢ α - 1 4 ⁢ π ) subscript 𝑔 𝑛 superscript 2 𝜋 𝛼 1 2 Euler-Gamma 𝑛 1 2 𝑛 𝑛 𝛼 1 2 𝛼 1 4 𝜋 {\displaystyle{\displaystyle g_{n}=\left(\frac{2}{\pi\sin\alpha}\right)^{1/2}% \frac{\Gamma\left(n+\frac{1}{2}\right)}{n!}\cos\left(n\alpha+\tfrac{1}{2}% \alpha-\tfrac{1}{4}\pi\right)}}
g_{n} = \left(\frac{2}{\pi\sin@@{\alpha}}\right)^{1/2}\frac{\EulerGamma@{n+\frac{1}{2}}}{n!}\cos@{n\alpha+\tfrac{1}{2}\alpha-\tfrac{1}{4}\pi}		 ℜ ⁡ ( n + 1 2 ) > 0 𝑛 1 2 0 {\displaystyle{\displaystyle\Re(n+\frac{1}{2})>0}} 
g[n] = ((2)/(Pi*sin(alpha)))^(1/2)*(GAMMA(n +(1)/(2)))/(factorial(n))*cos(n*alpha +(1)/(2)*alpha -(1)/(4)*Pi)

Subscript[g, n] == (Divide[2,Pi*Sin[\[Alpha]]])^(1/2)*Divide[Gamma[n +Divide[1,2]],(n)!]*Cos[n*\[Alpha]+Divide[1,2]*\[Alpha]-Divide[1,4]*Pi]
Failure Failure
Failed [90 / 90]
Result: .7909815655+.5000000000*I
Test Values: {alpha = 1.5, g[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.388725754+.5000000000*I
Test Values: {alpha = 1.5, g[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: .9745517365+.5000000000*I
Test Values: {alpha = 1.5, g[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -.5750438385+.8660254040*I
Test Values: {alpha = 1.5, g[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [90 / 90]
Result: Complex[0.7909815648537277, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[α, 1.5], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.3887257535176638, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[α, 1.5], Rule[Subscript[g, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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