rwg>try sum (csc π φ n)/n. Julian showed the largest terms have n = fib(k), and Csc[π*GoldenRatio*Fibonacci[k]]/Fibonacci[k] -> (-1)^Ceiling[k/3]*Sqrt[5]/π) so the partial sums of just the "spike" terms oscillate around 0 with period 6. So, what about csc(π k coth 1)/(k log k)? Mathematica startled me with a closed form ("obvious" in retrospect) for the convergents of coth 1: In[612]:= MatrixForm[Out[585]] Out[612]//MatrixForm= a[n] a[-1 + n] b[n] b[-1 + n] n 1 1 (-1) Pi BesselI[- + n, 1] + 2 E BesselK[- + n, 1] Cosh[1] 2 2 {a[n_] :> ----------------------------------------------------------, E Sqrt[2 Pi] n 1 1 -(-1) Pi BesselI[- + n, 1] + 2 E BesselK[- + n, 1] Sinh[1] 2 2 b[n_] :> -----------------------------------------------------------} E Sqrt[2 Pi] In[589]:= Table[MatrixForm[FullSimplify[FunctionExpand[%585 /. %588]]], {n, 0, 6}] Out[589]= {1 0, 1 1, 4 1, 21 4 , 151 21 , 1380 151 , 15331 1380 } 0 1 1 0 3 1 16 3 115 16 1051 115 11676 1051 In[594]:= Dot @@ Table[MatrixForm[{{2*k - 1, 1}, {1, 0}}], {k, 6}] Out[594]= 1 1 . 3 1 . 5 1 . 7 1 . 9 1 . 11 1 1 0 1 0 1 0 1 0 1 0 1 0 In[595]:= Thread[%, MatrixForm] Out[595]//MatrixForm= 15331 1380 11676 1051 In[571]:= Convergents[Coth[1], 6] Out[571]= {1, 4/3, 21/16, 151/115, 1380/1051, 15331/11676} whereupon Julian determined that the sequence {Csc[π*k*Coth[1]]/k} spikes when k=b[n], approaching (-1)^Floor[(n+2)/3]*(2*n+1)/π: Out[610]= Pi Csc[Pi b[n] Coth[1]] ------------------------------------- Floor[(2 + n)/3] (-1) ((1 + 2 n) b[n]) In[611]:= N[Table[N[% /. %588, 69], {n, 6, 9}]] Out[611]= {1.01203, 1.009, 1.00699, 1.00559} We have not yet examined the intervening partial sums. --rwg In a stunning breakthrough, Google has automated translation into Engrish, providing a bountiful new source of examples of this mellifluous language: 一方、3月31日も事態の安定に向けた努力が続く原発では、世界の原子力大国の注目が高まっている。 YouTubeに掲載された映像に映っていたのは、水平線から迫り来る大津波で、まさに発電所に牙をむこうとしていた。 On the other hand, in the primary followed by efforts to stabilize the situation also March 31, the attention of the world's nuclear power has been increasing. It was reflected in the video are posted to YouTube, looming large in the tsunami from the horizontal, son-in-law was trying to power plant just fangs .