Asymptotic Approximations - 2.11 Remainder Terms; Stokes Phenomenon

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
2.11.E1 I ⁢ ( m ) = ∫ 0 π cos ⁡ ( m ⁢ t ) t 2 + 1 ⁢ d t 𝐼 𝑚 superscript subscript 0 𝜋 𝑚 𝑡 superscript 𝑡 2 1 𝑡 {\displaystyle{\displaystyle I(m)=\int_{0}^{\pi}\frac{\cos\left(mt\right)}{t^{% 2}+1}\mathrm{d}t}}
I(m) = \int_{0}^{\pi}\frac{\cos@{mt}}{t^{2}+1}\diff{t}		 

I(m) = int((cos(m*t))/((t)^(2)+ 1), t = 0..Pi)

I[m] == Integrate[Divide[Cos[m*t],(t)^(2)+ 1], {t, 0, Pi}, GenerateConditions->None]
Failure Failure
Failed [30 / 30]
Result: 2.012164326+2.811364624*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, m = 1}


Result: 3.764776118+6.449767277*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, m = 2}


Result: 10.44871992+12.82571836*I
Test Values: {I = 1/2*3^(1/2)+1/2*I, m = 3}


Result: -.5451540752-.6604650959*I
Test Values: {I = -1/2+1/2*I*3^(1/2), m = 1}

... skip entries to safe data
Failed [9 / 9]
Result: Complex[3.1301272053762923, 2.7021954356714506]
Test Values: {Rule[Complex[0, 1], 1], Rule[m, 1]}


Result: Complex[7.946986696458338, 4.871470912282225]
Test Values: {Rule[Complex[0, 1], 1], Rule[m, 2]}

... skip entries to safe data
2.11#Ex1 q 1 ⁢ ( t ) = - 2 ⁢ t ( t 2 + 1 ) 2 subscript 𝑞 1 𝑡 2 𝑡 superscript superscript 𝑡 2 1 2 {\displaystyle{\displaystyle q_{1}(t)=-\frac{2t}{(t^{2}+1)^{2}}}}
q_{1}(t) = -\frac{2t}{(t^{2}+1)^{2}}		 

q[1](t) = -(2*t)/(((t)^(2)+ 1)^(2))
 
Subscript[q, 1][t] == -Divide[2*t,((t)^(2)+ 1)^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex2 q 2 ⁢ ( t ) = 24 ⁢ ( t 3 - t ) ( t 2 + 1 ) 4 subscript 𝑞 2 𝑡 24 superscript 𝑡 3 𝑡 superscript superscript 𝑡 2 1 4 {\displaystyle{\displaystyle q_{2}(t)=\frac{24(t^{3}-t)}{(t^{2}+1)^{4}}}}
q_{2}(t) = \frac{24(t^{3}-t)}{(t^{2}+1)^{4}}		 

q[2](t) = (24*((t)^(3)- t))/(((t)^(2)+ 1)^(4))
 
Subscript[q, 2][t] == Divide[24*((t)^(3)- t),((t)^(2)+ 1)^(4)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex3 q 3 ⁢ ( t ) = - 240 ⁢ ( 3 ⁢ t 5 - 10 ⁢ t 3 + 3 ⁢ t ) ( t 2 + 1 ) 6 subscript 𝑞 3 𝑡 240 3 superscript 𝑡 5 10 superscript 𝑡 3 3 𝑡 superscript superscript 𝑡 2 1 6 {\displaystyle{\displaystyle q_{3}(t)=-\frac{240(3t^{5}-10t^{3}+3t)}{(t^{2}+1)% ^{6}}}}
q_{3}(t) = -\frac{240(3t^{5}-10t^{3}+3t)}{(t^{2}+1)^{6}}		 

q[3](t) = -(240*(3*(t)^(5)- 10*(t)^(3)+ 3*t))/(((t)^(2)+ 1)^(6))
 
Subscript[q, 3][t] == -Divide[240*(3*(t)^(5)- 10*(t)^(3)+ 3*t),((t)^(2)+ 1)^(6)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E5 E p ⁡ ( z ) = e - z ⁢ z p - 1 Γ ⁡ ( p ) ⁢ ∫ 0 ∞ e - z ⁢ t ⁢ t p - 1 1 + t ⁢ d t exponential-integral-En 𝑝 𝑧 superscript 𝑒 𝑧 superscript 𝑧 𝑝 1 Euler-Gamma 𝑝 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑝 1 1 𝑡 𝑡 {\displaystyle{\displaystyle E_{p}\left(z\right)=\frac{e^{-z}z^{p-1}}{\Gamma% \left(p\right)}\int_{0}^{\infty}\frac{e^{-zt}t^{p-1}}{1+t}\mathrm{d}t}}
\genexpintE{p}@{z} = \frac{e^{-z}z^{p-1}}{\EulerGamma@{p}}\int_{0}^{\infty}\frac{e^{-zt}t^{p-1}}{1+t}\diff{t}		 ℜ ⁡ p > 0 𝑝 0 {\displaystyle{\displaystyle\Re p>0}} 
Ei(p, z) = (exp(- z)*(z)^(p - 1))/(GAMMA(p))*int((exp(- z*t)*(t)^(p - 1))/(1 + t), t = 0..infinity)

ExpIntegralE[p, z] == Divide[Exp[- z]*(z)^(p - 1),Gamma[p]]*Integrate[Divide[Exp[- z*t]*(t)^(p - 1),1 + t], {t, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 35]
2.11.E8 n = ρ - p + α 𝑛 𝜌 𝑝 𝛼 {\displaystyle{\displaystyle n=\rho-p+\alpha}}
n = \rho-p+\alpha		 

n = rho - p + alpha
 
n == \[Rho]- p + \[Alpha]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E9 1 1 + t = ∑ s = 0 n - 1 ( - 1 ) s ⁢ t s + ( - 1 ) n ⁢ t n 1 + t 1 1 𝑡 superscript subscript 𝑠 0 𝑛 1 superscript 1 𝑠 superscript 𝑡 𝑠 superscript 1 𝑛 superscript 𝑡 𝑛 1 𝑡 {\displaystyle{\displaystyle\frac{1}{1+t}=\sum_{s=0}^{n-1}(-1)^{s}t^{s}+(-1)^{% n}\frac{t^{n}}{1+t}}}
\frac{1}{1+t} = \sum_{s=0}^{n-1}(-1)^{s}t^{s}+(-1)^{n}\frac{t^{n}}{1+t}		 

(1)/(1 + t) = sum((- 1)^(s)* (t)^(s), s = 0..n - 1)+(- 1)^(n)*((t)^(n))/(1 + t)
 
Divide[1,1 + t] == Sum[(- 1)^(s)* (t)^(s), {s, 0, n - 1}, GenerateConditions->None]+(- 1)^(n)*Divide[(t)^(n),1 + t]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E11 e - z 2 ⁢ π ⁢ ∫ 0 ∞ e - z ⁢ t ⁢ t n + p - 1 1 + t ⁢ d t = Γ ⁡ ( n + p ) 2 ⁢ π ⁢ E n + p ⁡ ( z ) z n + p - 1 superscript 𝑒 𝑧 2 𝜋 superscript subscript 0 superscript 𝑒 𝑧 𝑡 superscript 𝑡 𝑛 𝑝 1 1 𝑡 𝑡 Euler-Gamma 𝑛 𝑝 2 𝜋 exponential-integral-En 𝑛 𝑝 𝑧 superscript 𝑧 𝑛 𝑝 1 {\displaystyle{\displaystyle\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t% ^{n+p-1}}{1+t}\mathrm{d}t=\frac{\Gamma\left(n+p\right)}{2\pi}\frac{E_{n+p}% \left(z\right)}{z^{n+p-1}}}}
\frac{e^{-z}}{2\pi}\int_{0}^{\infty}\frac{e^{-zt}t^{n+p-1}}{1+t}\diff{t} = \frac{\EulerGamma@{n+p}}{2\pi}\frac{\genexpintE{n+p}@{z}}{z^{n+p-1}}		 ℜ ⁡ ( n + p ) > 0 𝑛 𝑝 0 {\displaystyle{\displaystyle\Re(n+p)>0}} 
(exp(- z))/(2*Pi)*int((exp(- z*t)*(t)^(n + p - 1))/(1 + t), t = 0..infinity) = (GAMMA(n + p))/(2*Pi)*(Ei(n + p, z))/((z)^(n + p - 1))

Divide[Exp[- z],2*Pi]*Integrate[Divide[Exp[- z*t]*(t)^(n + p - 1),1 + t], {t, 0, Infinity}, GenerateConditions->None] == Divide[Gamma[n + p],2*Pi]*Divide[ExpIntegralE[n + p, z],(z)^(n + p - 1)]
Successful Aborted - Successful [Tested: 189]
2.11.E14 a 2 ⁢ ( θ , α ) = 1 12 ⁢ ( 6 ⁢ α 2 - 6 ⁢ α + 1 ) - α 1 + e i ⁢ θ + 1 ( 1 + e i ⁢ θ ) 2 subscript 𝑎 2 𝜃 𝛼 1 12 6 superscript 𝛼 2 6 𝛼 1 𝛼 1 superscript 𝑒 𝑖 𝜃 1 superscript 1 superscript 𝑒 𝑖 𝜃 2 {\displaystyle{\displaystyle a_{2}(\theta,\alpha)=\frac{1}{12}(6\alpha^{2}-6% \alpha+1)-\frac{\alpha}{1+e^{i\theta}}+\frac{1}{(1+e^{i\theta})^{2}}}}
a_{2}(\theta,\alpha) = \frac{1}{12}(6\alpha^{2}-6\alpha+1)-\frac{\alpha}{1+e^{i\theta}}+\frac{1}{(1+e^{i\theta})^{2}}		 

a[2](theta , alpha) = (1)/(12)*(6*(alpha)^(2)- 6*alpha + 1)-(alpha)/(1 + exp(I*theta))+(1)/((1 + exp(I*theta))^(2))
 
Subscript[a, 2][\[Theta], \[Alpha]] == Divide[1,12]*(6*\[Alpha]^(2)- 6*\[Alpha]+ 1)-Divide[\[Alpha],1 + Exp[I*\[Theta]]]+Divide[1,(1 + Exp[I*\[Theta]])^(2)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E16 c ⁢ ( θ ) = 2 ⁢ ( 1 + e i ⁢ θ + i ⁢ ( θ - π ) ) 𝑐 𝜃 2 1 superscript 𝑒 𝑖 𝜃 𝑖 𝜃 𝜋 {\displaystyle{\displaystyle c(\theta)=\sqrt{2(1+e^{i\theta}+i(\theta-\pi))}}}
c(\theta) = \sqrt{2(1+e^{i\theta}+i(\theta-\pi))}		 

c(theta) = sqrt(2*(1 + exp(I*theta)+ I*(theta - Pi)))
 
c[\[Theta]] == Sqrt[2*(1 + Exp[I*\[Theta]]+ I*(\[Theta]- Pi))]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E17 h 2 ⁢ s ⁢ ( θ , α ) = e i ⁢ α ⁢ ( π - θ ) 1 + e - i ⁢ θ ⁢ a 2 ⁢ s ⁢ ( θ , α ) + ( - 1 ) s - 1 ⁢ i ⁢ 1 ⋅ 3 ⋅ 5 ⁢ ⋯ ⁢ ( 2 ⁢ s - 1 ) ( c ⁢ ( θ ) ) 2 ⁢ s + 1 subscript ℎ 2 𝑠 𝜃 𝛼 superscript 𝑒 𝑖 𝛼 𝜋 𝜃 1 superscript 𝑒 𝑖 𝜃 subscript 𝑎 2 𝑠 𝜃 𝛼 superscript 1 𝑠 1 𝑖 ⋅ 1 3 5 ⋯ 2 𝑠 1 superscript 𝑐 𝜃 2 𝑠 1 {\displaystyle{\displaystyle h_{2s}(\theta,\alpha)=\frac{e^{i\alpha(\pi-\theta% )}}{1+e^{-i\theta}}a_{2s}(\theta,\alpha)+(-1)^{s-1}i\frac{1\cdot 3\cdot 5\cdot% \cdot\cdot(2s-1)}{(c(\theta))^{2s+1}}}}
h_{2s}(\theta,\alpha) = \frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}a_{2s}(\theta,\alpha)+(-1)^{s-1}i\frac{1\cdot 3\cdot 5\cdot\cdot\cdot(2s-1)}{(c(\theta))^{2s+1}}		 

h[2*s](theta , alpha) = (exp(I*alpha*(Pi - theta)))/(1 + exp(- I*theta))*a[2*s](theta , alpha)+(- 1)^(s - 1)* I*(1 * 3 * 5 * * *(2*s - 1))/((c(theta))^(2*s + 1))
 
Subscript[h, 2*s][\[Theta], \[Alpha]] == Divide[Exp[I*\[Alpha]*(Pi - \[Theta])],1 + Exp[- I*\[Theta]]]*Subscript[a, 2*s][\[Theta], \[Alpha]]+(- 1)^(s - 1)* I*Divide[1 * 3 * 5 * * *(2*s - 1),(c[\[Theta]])^(2*s + 1)]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E18 h 0 ⁢ ( θ , α ) = e i ⁢ α ⁢ ( π - θ ) 1 + e - i ⁢ θ - i c ⁢ ( θ ) subscript ℎ 0 𝜃 𝛼 superscript 𝑒 𝑖 𝛼 𝜋 𝜃 1 superscript 𝑒 𝑖 𝜃 𝑖 𝑐 𝜃 {\displaystyle{\displaystyle h_{0}(\theta,\alpha)=\frac{e^{i\alpha(\pi-\theta)% }}{1+e^{-i\theta}}-\frac{i}{c(\theta)}}}
h_{0}(\theta,\alpha) = \frac{e^{i\alpha(\pi-\theta)}}{1+e^{-i\theta}}-\frac{i}{c(\theta)}		 

h[0](theta , alpha) = (exp(I*alpha*(Pi - theta)))/(1 + exp(- I*theta))-(I)/(c(theta))
 
Subscript[h, 0][\[Theta], \[Alpha]] == Divide[Exp[I*\[Alpha]*(Pi - \[Theta])],1 + Exp[- I*\[Theta]]]-Divide[I,c[\[Theta]]]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E19 w j ⁢ ( z ) = e λ j ⁢ z ⁢ z μ j ⁢ ∑ s = 0 n - 1 a s , j z s + R n ( j ) ⁢ ( z ) subscript 𝑤 𝑗 𝑧 superscript 𝑒 subscript 𝜆 𝑗 𝑧 superscript 𝑧 subscript 𝜇 𝑗 superscript subscript 𝑠 0 𝑛 1 subscript 𝑎 𝑠 𝑗 superscript 𝑧 𝑠 superscript subscript 𝑅 𝑛 𝑗 𝑧 {\displaystyle{\displaystyle w_{j}(z)=e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{n% -1}\frac{a_{s,j}}{z^{s}}+R_{n}^{(j)}(z)}}
w_{j}(z) = e^{\lambda_{j}z}z^{\mu_{j}}\sum_{s=0}^{n-1}\frac{a_{s,j}}{z^{s}}+R_{n}^{(j)}(z)		 

w[j](z) = exp(lambda[j]*z)*(z)^(mu[j])* sum((a[s , j])/((z)^(s)), s = 0..n - 1)+ (R[n])^(j)(z)
 
Subscript[w, j][z] == Exp[Subscript[\[Lambda], j]*z]*(z)^(Subscript[\[Mu], j])* Sum[Divide[Subscript[a, s , j],(z)^(s)], {s, 0, n - 1}, GenerateConditions->None]+ (Subscript[R, n])^(j)[z]
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E26 e 5 ⁢ E 1 ⁡ ( 5 ) = 0.17042 ⁢ … superscript 𝑒 5 exponential-integral 5 0.17042 … {\displaystyle{\displaystyle e^{5}E_{1}\left(5\right)=0.17042\dots}}
e^{5}\expintE@{5} = 0.17042\dots		 

exp(5)*Ei(5) = 0.17042

Exp[5]*ExpIntegralE[1, 5] == 0.17042
Failure Failure Skip - No test values generated Successful [Tested: 1]
2.11#Ex4 Δ 0 = 0.00768 superscript Δ 0 0.00768 {\displaystyle{\displaystyle\Delta^{0}=0.00768}}
\Delta^{0} = 0.00768		 

(Delta)^(0) = 0.00768
 
\[CapitalDelta]^(0) == 0.00768
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex5 Δ 1 = 0.00154 superscript Δ 1 0.00154 {\displaystyle{\displaystyle\Delta^{1}=0.00154}}
\Delta^{1} = 0.00154		 

(Delta)^(1) = 0.00154
 
\[CapitalDelta]^(1) == 0.00154
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex6 Δ 2 = 0.00214 superscript Δ 2 0.00214 {\displaystyle{\displaystyle\Delta^{2}=0.00214}}
\Delta^{2} = 0.00214		 

(Delta)^(2) = 0.00214
 
\[CapitalDelta]^(2) == 0.00214
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex7 Δ 3 = 0.00192 superscript Δ 3 0.00192 {\displaystyle{\displaystyle\Delta^{3}=0.00192}}
\Delta^{3} = 0.00192		 

(Delta)^(3) = 0.00192
 
\[CapitalDelta]^(3) == 0.00192
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex8 Δ 4 = 0.00280 superscript Δ 4 0.00280 {\displaystyle{\displaystyle\Delta^{4}=0.00280}}
\Delta^{4} = 0.00280		 

(Delta)^(4) = 0.00280
 
\[CapitalDelta]^(4) == 0.00280
 Skipped - no semantic math Skipped - no semantic math - -
2.11#Ex9 Δ 5 = 0.00434 superscript Δ 5 0.00434 {\displaystyle{\displaystyle\Delta^{5}=0.00434}}
\Delta^{5} = 0.00434		 

(Delta)^(5) = 0.00434
 
\[CapitalDelta]^(5) == 0.00434
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E28 0.00384 - 0.00038 + 0.00027 - 0.00012 + 0.00009 - 0.00007 = 0.00363 0.00384 0.00038 0.00027 0.00012 0.00009 0.00007 0.00363 {\displaystyle{\displaystyle 0.00384-0.00038+0.00027-0.00012+0.00009-0.00007=0% .00363}}
0.00384-0.00038+0.00027-0.00012+0.00009-0.00007 = 0.00363		 

0.00384 - 0.00038 + 0.00027 - 0.00012 + 0.00009 - 0.00007 = 0.00363
 
0.00384 - 0.00038 + 0.00027 - 0.00012 + 0.00009 - 0.00007 == 0.00363
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E30 a n = e - z / 2 z n - κ ⁢ n ! ⁢ ( μ 2 - ( κ - 1 2 ) 2 ) ⁢ ( μ 2 - ( κ - 3 2 ) 2 ) ⁢ ⋯ ⁢ ( μ 2 - ( κ - n + 1 2 ) 2 ) subscript 𝑎 𝑛 superscript 𝑒 𝑧 2 superscript 𝑧 𝑛 𝜅 𝑛 superscript 𝜇 2 superscript 𝜅 1 2 2 superscript 𝜇 2 superscript 𝜅 3 2 2 ⋯ superscript 𝜇 2 superscript 𝜅 𝑛 1 2 2 {\displaystyle{\displaystyle a_{n}=\frac{e^{-z/2}}{z^{n-\kappa}n!}\left(\mu^{2% }-(\kappa-\tfrac{1}{2})^{2}\right)\*\left(\mu^{2}-(\kappa-\tfrac{3}{2})^{2}% \right)\*\cdot\cdot\cdot\left(\mu^{2}-(\kappa-n+\tfrac{1}{2})^{2}\right)}}
a_{n} = \frac{e^{-z/2}}{z^{n-\kappa}n!}\left(\mu^{2}-(\kappa-\tfrac{1}{2})^{2}\right)\*\left(\mu^{2}-(\kappa-\tfrac{3}{2})^{2}\right)\*\cdot\cdot\cdot\left(\mu^{2}-(\kappa-n+\tfrac{1}{2})^{2}\right)		 

a[n] = (exp(- z/2))/((z)^(n - kappa)* factorial(n))*((mu)^(2)-(kappa -(1)/(2))^(2))*((mu)^(2)-(kappa -(3)/(2))^(2))* * * *((mu)^(2)-(kappa - n +(1)/(2))^(2))
 
Subscript[a, n] == Divide[Exp[- z/2],(z)^(n - \[Kappa])* (n)!]*(\[Mu]^(2)-(\[Kappa]-Divide[1,2])^(2))*(\[Mu]^(2)-(\[Kappa]-Divide[3,2])^(2))* * * *(\[Mu]^(2)-(\[Kappa]- n +Divide[1,2])^(2))
 Skipped - no semantic math Skipped - no semantic math - -
2.11.E31 W 2.3 , 0.5 ⁡ ( 1.0 ) = - 0.83299 50268 27526 ⁢ ⋯ Whittaker-confluent-hypergeometric-W 2.3 0.5 1.0 0.83299 50268 27526 ⋯ {\displaystyle{\displaystyle W_{2.3,0.5}\left(1.0\right)=-0.83299\;50268\;2752% 6\;\cdots}}
\WhittakerconfhyperW{2.3}{0.5}@{1.0} = -0.83299\;50268\;27526\;\cdots		 

WhittakerW(2.3, 0.5, 1.0) = - 0.832995026827526

WhittakerW[2.3, 0.5, 1.0] == - 0.832995026827526
Successful Failure - Successful [Tested: 1]
2.11.E32 d n = ∑ j = 0 n ( - 1 ) j ⁢ ( n j ) ⁢ ( j + 1 ) n - 1 ⁢ s j a j + 1 ∑ j = 0 n ( - 1 ) j ⁢ ( n j ) ⁢ ( j + 1 ) n - 1 ⁢ 1 a j + 1 subscript 𝑑 𝑛 superscript subscript 𝑗 0 𝑛 superscript 1 𝑗 binomial 𝑛 𝑗 superscript 𝑗 1 𝑛 1 subscript 𝑠 𝑗 subscript 𝑎 𝑗 1 superscript subscript 𝑗 0 𝑛 superscript 1 𝑗 binomial 𝑛 𝑗 superscript 𝑗 1 𝑛 1 1 subscript 𝑎 𝑗 1 {\displaystyle{\displaystyle d_{n}=\frac{\sum_{j=0}^{n}(-1)^{j}\genfrac{(}{)}{% 0.0pt}{}{n}{j}(j+1)^{n-1}\frac{s_{j}}{a_{j+1}}}{\sum_{j=0}^{n}(-1)^{j}\genfrac% {(}{)}{0.0pt}{}{n}{j}(j+1)^{n-1}\frac{1}{a_{j+1}}}}}
d_{n} = \frac{\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(j+1)^{n-1}\frac{s_{j}}{a_{j+1}}}{\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}(j+1)^{n-1}\frac{1}{a_{j+1}}}		 

d[n] = (sum((- 1)^(j)*binomial(n,j)*(j + 1)^(n - 1)*(s[j])/(a[j + 1]), j = 0..n))/(sum((- 1)^(j)*binomial(n,j)*(j + 1)^(n - 1)*(1)/(a[j + 1]), j = 0..n))

Subscript[d, n] == Divide[Sum[(- 1)^(j)*Binomial[n,j]*(j + 1)^(n - 1)*Divide[Subscript[s, j],Subscript[a, j + 1]], {j, 0, n}, GenerateConditions->None],Sum[(- 1)^(j)*Binomial[n,j]*(j + 1)^(n - 1)*Divide[1,Subscript[a, j + 1]], {j, 0, n}, GenerateConditions->None]]
Failure Failure Error Skipped - Because timed out
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