Find Closed Forms for Number Theoretic Sums and Products
A sampling of sums and products.
 In[1]:= ```problems = {HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((3\ k + 1)\), \(3\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)] \*FractionBox[\(Sin[2\ \[Pi]\ k\ x]\), SqrtBox[\(k\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = \(-\[Infinity]\)\), \(\ \[Infinity]\)]\( \*SuperscriptBox[\((\(-1\))\), \(k\)]\ \*SuperscriptBox[\(q\), SuperscriptBox[\(k\), \(2\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(\[Infinity]\)] \*FractionBox[ SuperscriptBox[\((2\ x)\), \(2\ m + 2\ k\)], \( \*SuperscriptBox[\(m\), \(2\)]\ \((m + k)\)\ Binomial[2\ m, m]\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), \(Floor[k/5 + Sqrt[7]]^2\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(\[Infinity]\)]\((1 + \*FractionBox[\(1\), SuperscriptBox[\(k\), \(2\)]])\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Product]\), \(k = 0\), \(\[Infinity]\)]\((1 - a\ \*SuperscriptBox[\(q\), \(k\)])\)\)], HoldForm[ Product[1 + 1/(k Floor[(k^2 + 4)/(k + 1)]), {k, 1, \[Infinity]}]]};```
 In[2]:= ```FormulaGallery[forms_List] := Module[{vals = ParallelMap[ReleaseHold, forms]}, Text@TraditionalForm@ Grid[Table[{forms[[i]], "==", vals[[i]]}, {i, Length[forms]}], Dividers -> {{True, False, False, True}, All}, Alignment -> {{Right, Center, Left}, Baseline}, Background -> LightYellow, Spacings -> {1, 1}]];```
 In[3]:= `FormulaGallery[problems]`
 Out[3]=