[math-fun] Strange-ish dilog sum
Sum[(-Log[2])*Log[x] + 2^n*(Pi^2/12 + PolyLog[2, -x^(-2^(-n))]), {n, 0, Infinity}] -> -(Pi^2/12) + Log[x] - Log[x]*Log[2*Log[x]] + (1/2)*PolyLog[2, 1/x^2] Doesn't look too convergent, does it? Mma doubts it too. But In[389]:= MapAt[22 &, %, {1, 2, 3}] /. x -> .69 Out[389]= -0.0716747 -> -0.0716748 - 2.22045*10^-16 I I think there are 3^-n versions of this as well. --rwg
Foo, this is just the k=2 case of the telescoping identity n-1 li_k(-u^2^j) li_k(u^2^n) sum ------------ = ------------ - li_k(u) . j=0 2^((k-1)j) 2^((k-1)n) The u^3^n variants are almost as trivial. --rwg IMPERSONATE PERMEATIONS IMPERSONATED PREDOMINATES IMPERSONATING IMPREGNATIONS
Sum[(-Log[2])*Log[x] + 2^n*(Pi^2/12 + PolyLog[2, -x^(-2^(-n))]), {n, 0, Infinity}] -> -(Pi^2/12) + Log[x] - Log[x]*Log[2*Log[x]] + (1/2)*PolyLog[2, 1/x^2]
Doesn't look too convergent, does it? Mma doubts it too. But In[389]:= MapAt[22 &, %, {1, 2, 3}] /. x -> .69
Out[389]= -0.0716747 -> -0.0716748 - 2.22045*10^-16 I I think there are 3^-n versions of this as well. --rwg
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Bill Gosper -
rwg＠sdf.lonestar.org