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I try this numerical summation (in two parts)

a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}, WorkingPrecision -> 100, PrecisionGoal -> 100]; N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100] 

and this numerical integration

NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, WorkingPrecision -> 100, PrecisionGoal -> 100] 

which are supposed to give the same result, and they do, but only to 25 places.

Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}] 

1/144 (π^4+72 (EulerGamma + Log[4]) Zeta[3] - 36 Sqrt[π] 

 (HypergeometricPFQRegularized^({0, 0, 0, 0, 0}, {0, 0, 0, 1}, 0))[ {1, 1, 1,1, 3/2}, {2, 2, 2, 3/2}, 1]) 

However, N[b,20] never returns. The problem seems to be the evaluation of the derivative of HypergeometricPFQRegularized.

I try this numerical summation (in two parts)

a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}, WorkingPrecision -> 100, PrecisionGoal -> 100]; N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100] 

and this numerical integration

NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, WorkingPrecision -> 100, PrecisionGoal -> 100] 

which are supposed to give the same result, and they do, but only to 25 places.

Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}] 

1/144 (π^4 + 72 (EulerGamma + Log[4]) Zeta[3] - 36 Sqrt[π] 

 (HypergeometricPFQRegularized^({0, 0, 0, 0, 0}, {0, 0, 0, 1}, 0))[ {1, 1, 1,1, 3/2}, {2, 2, 2, 3/2}, 1]) 

However, N[b,20] never returns. The problem seems to be the evaluation of the derivative of HypergeometricPFQRegularized.

I try this numerical summation (in two parts)

a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}, WorkingPrecision -> 100, PrecisionGoal -> 100]; N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100] 

and this numerical integration

NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2},   WorkingPrecision -> 100, PrecisionGoal -> 100] 

which are supposed to give the same result, and they do, but only to 25 places.

Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}]   (* 1/144 (π^4+72 (EulerGamma+Log[4]) Zeta[3]-36 Sqrt[π]  (HypergeometricPFQRegularized^({0,0,0,0,0},{0,0,0,1},0))[{1,1,1,1,3/2},{2,2,2,3/2},1]) *) 

however, N[b,20] never returns. The problem seems to be the evaluation of the derivative of HypergeometricPFQRegularized.

I try this numerical summation (in two parts)

a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity},   WorkingPrecision -> 100, PrecisionGoal -> 100]; N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100] 

and this numerical integration

NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, WorkingPrecision -> 100, PrecisionGoal -> 100] 

which are supposed to give the same result, and they do, but only to 25 places.

Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}] 

1/144 (π^4+72 (EulerGamma + Log[4]) Zeta[3] - 36 Sqrt[π] 

 (HypergeometricPFQRegularized^({0, 0, 0, 0, 0}, {0, 0, 0, 1}, 0))[ {1, 1, 1,1, 3/2}, {2, 2, 2, 3/2}, 1]) 

However, N[b,20] never returns. The problem seems to be the evaluation of the derivative of HypergeometricPFQRegularized.

I try this numerical summation (in two parts)

a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}, WorkingPrecision -> 100, PrecisionGoal -> 100]

N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100]

and this numerical integration

NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, WorkingPrecision -> 100, PrecisionGoal -> 100]

which are supposed to give the same result, and they do, but only to 25 places. Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}]

1/144 (\[Pi]^4+72 (EulerGamma+Log[4]) Zeta[3]-36 Sqrt[\[Pi]] (HypergeometricPFQRegularized^({0,0,0,0,0},{0,0,0,1},0))[{1,1,1,1,3/2},{2,2,2,3/2},1])

however, N[b,20] never returns. The problem seems to be the evaluation of the derivative of HypergeometricPFQRegularized.

I try this numerical summation (in two parts)

a = NSum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}, WorkingPrecision -> 100, PrecisionGoal -> 100]; N[Pi^3/24 Log[2]^2 + Log[2] Pi/16 Zeta[3] - Pi^5/960 - Pi/16 a, 100] 

and this numerical integration

NIntegrate[x^2 Log[Sin[x]] Log[Cos[x]], {x, 0, Pi/2}, WorkingPrecision -> 100, PrecisionGoal -> 100] 

which are supposed to give the same result, and they do, but only to 25 places. 

Obviously, at least one of the results is off. How can I increase the precision so that these agree past 25 places? If that can't be done, which of these is the more accurate?

If I try to evaluate the first quantity symbolically, I get

b = Sum[(HarmonicNumber[2 m])/m^3, {m, 1, Infinity}] (* 1/144 (π^4+72 (EulerGamma+Log[4]) Zeta[3]-36 Sqrt[π] (HypergeometricPFQRegularized^({0,0,0,0,0},{0,0,0,1},0))[{1,1,1,1,3/2},{2,2,2,3/2},1]) *) 

however, N[b,20] never returns. The problem seems to be the evaluation of the derivative of HypergeometricPFQRegularized.