Compute a Collection of Definite Sums
Compute a collection of definite sums, of finite and infinite domains.
 In[1]:= ```problems = {HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(m\)] \*FractionBox[\(5\ \*SuperscriptBox[\(q\), \(k\)]\), \(\((1 - 5\ \*SuperscriptBox[\(q\), \(k\)])\)\ QPochhammer[5, q, k]\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(Floor[ \*FractionBox[\(n\), \(2\)]]\)] \*FractionBox[\( \*SuperscriptBox[\((\(-1\))\), \(k\)]\ \*SuperscriptBox[\(2\), \(n - 2\ k\)]\ Binomial[n - k, k]\), \(n + 1\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(q - 1\)]\(Cos[ \*FractionBox[\(4\ \[Pi]\ k\), \(q\)]]\ \*SuperscriptBox[\(Csc[ \*FractionBox[\(\[Pi]\ k\), \(q\)]]\), \(4\)]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(m\)]\(Binomial[n, k]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(n\)]\(\(k!\)\ \*SuperscriptBox[\((\(-1\))\), \(n - k\)]\ Binomial[n, k]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(m\)] \*FractionBox[ SuperscriptBox[\(4\), \(k\)], \(Binomial[2\ k, k]\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)] \*FractionBox[\(HermiteH[2\ k + 1, x]\), \(\((2\ k + 1)\)!\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(3\ k + 5\), SuperscriptBox[\((2\ k - 1)\), \(8\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)] \*FractionBox[\(\(\((3\ k + 1)\)!\)\ \*SuperscriptBox[\(x\), \(k\)]\), \(\((5\ k + 2)\)!\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(PolyGamma[k]\), SuperscriptBox[\((k + 1)\), \(2\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(HarmonicNumber[k, 2]\), SuperscriptBox[\(k\), \(2\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)]\(ArcTan[ \*FractionBox[\(2\ \*SuperscriptBox[\(3\), \(k\)]\), \(2 + \*SuperscriptBox[\(3\), \(k\)] + \*SuperscriptBox[\(3\), \(1 + k\)] + \*SuperscriptBox[\(3\), \(1 + 2\ k\)]\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(LucasL[\ k]\), SuperscriptBox[\(2\), \(k\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 2\), \(\[Infinity]\)] \*FractionBox[\(\(\ \)\(Log[k]\)\), SuperscriptBox[\(E\), \(k\)]]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)] \*FractionBox[\(StirlingS2[k + 1, k]\), \( \*SuperscriptBox[\(5\), \(k\)]\ \(k!\)\)]\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)]\(k\ \*SuperscriptBox[\(BesselJ[k, z]\), \(2\)]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 2\), \(\[Infinity]\)] \*FractionBox[\(Zeta[2\ k]\), \(\((2\ k + 1)\)\ \((2\ k + 2)\)\ \*SuperscriptBox[\(2\), \(2\ k\)]\)]\)], HoldForm[Sum[1/Mod[k^2, k^3, k^4], {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 -> {{4, {}, 4}, 1}]]```
 In[3]:= `FormulaGallery[problems]`
 Out[3]=