[math-fun] peculiar log π series

For positive integer k, Out[77]= Log[Pi] == EulerGamma + 2 Sum[Beta[1/(2*n), 1 + k, 0], {n, Infinity}] + Sum[(2^(1 - j)*Zeta[j])/j, {j, 2, k}] In[78]:= Table[%, {k, 4}] Out[78]= {True, True, Log[Pi] == EulerGamma + Pi^2/24 + Sum[2*Beta[1/(2*n), 4, 0], {n, Infinity}] + Zeta[3]/12, Log[Pi] == EulerGamma + Pi^2/24 + Pi^4/2880 + Sum[2*Beta[1/(2*n), 5, 0], {n, Infinity}] + Zeta[3]/12} where Beta[1/2/n,s,0] is In[83]:= Integrate[t^(s - 1)/(1 - t), {t, 0, 1/2/n}] Out[83]= ConditionalExpression[Beta[1/(2*n), s, 0], (2*Re[n] >= 1 || Re[n] <= 0 || NotElement[n, Reals]) && Re[s] > 0] || := Or. —rwg