Interestingly, Maple gives the equivalent, after assuming x is real, of arcsin(1/sqrt(x^4+1)) + arcsin(x^2/sqrt(x^4+1)). It also can’t simplify this to Pi/2. By hand it is of course straightforward to use arcsin(x)=arctan(x/sqrt(1-x^2)) and arctan(u)+arctan(v) = arctan((u+v)/(1-uv)) and simplify to arctan(Infinity)=Pi/2. Looks like both Maple and Mathematica are unable to simplify a sum of arcsines like this one. Steve On Dec 16, 2019, at 11:59 AM, Bill Gosper <billgosper@gmail.com<mailto:billgosper@gmail.com>> wrote: Hi François, Mathematica immediately gives In[152]:= Assuming[x \[Element] Reals, FullSimplify[Pi/2 == Sum[(2*n)!*(1 + x^(4*n + 2))/(2^(2*n)*(n!)^2*(1 + x^4)^(n + 1/2)*(2*n + 1)), {n, 0, \[Infinity]}]]] Out[152]= 2 (ArcCsc[Sqrt[1 + x^4]] + ArcSin[x^2/Sqrt[1 + x^4]]) == \[Pi] but is unable to finish the proof, nor even plot the constant. You have apparently exposed another headache for the developers. —rwg On Mon, Dec 16, 2019 at 6:29 AM françois mendzina essomba2 <m_essob@yahoo.fr<mailto:m_essob@yahoo.fr>> wrote: Hi, Pi/2==sum((2*n)!*(1+x^(4*n+2))/(2^(2*n)*(n!)^2*(1+x^4)^(n+1/2)*(2*n+1)) ,n=0..inf); an identity transformation gives this result for any real number. I wonder what other process can lead to this formula Best regards. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...