Explore Multivariate Sums
Explore definite multivariate sums.
 In[1]:= problems = {HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] FractionBox[ SuperscriptBox[\((\(-1\))\), \(m + n\)], SuperscriptBox[\((m + n)\), \(2\)]] m\ n\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 0\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)]\ \ FractionBox[\(\((m + n)\)!\), \(\((\(m!\))\)\ \((\(n!\))\)\)] \*SuperscriptBox[\(( \*FractionBox[\(x\), \(2\)])\), \(m + n\)]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 0\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)] FractionBox[\(\((m + n + k)\)!\), \(\((\(m!\))\) \((\(n!\))\) \((\(k!\))\)\)] \*SuperscriptBox[\(( \*FractionBox[\(x\), \(3\)])\), \(m + n + k\)]\)\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 2\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 2\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((a + n)\), \(m\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 2\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((2 n)\), \(m\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((4 n - 1)\), \(2 m\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((4 n - 1)\), \(2 m + 1\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((4 n - 2)\), \(2 m\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[\( SuperscriptBox[\(i\), \(2\)] j\), \( SuperscriptBox[\(3\), \(i\)] \((j\ \*SuperscriptBox[\(3\), \(i\)] + \ i\ \*SuperscriptBox[\(3\), \(j\)])\)\)]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = \(-\[Infinity]\)\), \(\ \[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = \(-\[Infinity]\)\), \(\ \[Infinity]\)] FractionBox[\(1\), SuperscriptBox[\(( \*SuperscriptBox[\(i\), \(2\)] + \*SuperscriptBox[\(j\), \(2\)])\), \(s\)]] Boole[{i, j} != {0, 0}]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((i\ j\ k)\), \(2\)]]\)\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), \( SuperscriptBox[\(2\), \(i\)] SuperscriptBox[\(2\), \(j\)] \*SuperscriptBox[\(2\), \(k\)]\)]\)\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[ SuperscriptBox[\((\(-1\))\), \(i + j\)], SuperscriptBox[\((i + j)\), \(3\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((i + j)\), \(3\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[\(\(\ \)\(Zeta[\ i + j]\)\), SuperscriptBox[\(2\), \(i + j\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), \(\((i + j)\)!\)]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), \(Max[i, j]!\)]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\(Max[i, j]\), \(3\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((i\ j + j\ k)\), \(s\)]]\)\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(\[Infinity]\)]\ \( \*UnderoverscriptBox[\(\[Sum]\), \(n = 1\), \(\[Infinity]\)] \*FractionBox[\(Zeta[m + 2 n]\), SuperscriptBox[\(4\), \( FractionBox[\(1\), \(2\)] m + n\)]]\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((i*j + j*k)\), \(s\)]]\)\)\)], HoldForm[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 0\), \(\[Infinity]\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(\[Infinity]\)] \*FractionBox[\(1\), SuperscriptBox[\((i + 2 j + k)\), \(4\)]]\)\)\)]};
 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]=