[math-fun] Sum csc(π k coth 1)/(k log k) converges? [Was: Teacher trolling]

18 May 2012

      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
.

[math-fun] Sum csc(π k coth 1)/(k log k) converges? [Was: Teacher trolling]

Bill Gosper