Bisecting the sum, plus minor fooling around in Mma gave Out[446]= LogIntegral[x] == CoshIntegral[Log[x]] + SinhIntegral[Log[x]] In[457]:= FullSimplify[%446, x > 1] Out[457]= True WDS> Ramanujan [notebook]: Li(x) = EulerMascheroniGamma + lnln(x) - sqrt(x) * SUM(n>0) a[n] * ln(x)^n where a[n] = (-1/2)^n / (n!) * SUM(0<=m<=floor((n-1)/2)) 2/(2*m+1) which I suppose has a nicer matrix product reformulation. <WDS Triangular 3x3, confluent. I seem to recall the nonexistence of confluent 2x2 path-invariant systems (except CF), but maybe luck is better with 3x3. If so, we'd get identities, but of what use? Even if the new matrices are triangular, it's unlikely they'd reveal any big surprises. And since Ramanujan's series is already confluent, there isn't much need for convergence speedup. --rwg