Expand series unevaluated

Question

I'm newbie in Mathematica. I'd like to obtain nice and verbose output for any series calculation.

For example, given a simple sequence n*(-1)^(n-1) and hypothetical function NoEval, I see it as follows:

sequence = n*(-1)^(n-1);
range = {n, 1, 10};
seriesFormula = NoEval[Sum[sequence, range]];
seriesExpanded = Sum[NoEval[sequence], range];
seriesSum = Sum[sequence, range];
Row[{seriesFormula, seriesExpanded, seriesSum}, "="]

expected output is:

Is there a simple way to do this?

I know there are Hold*, Unevaluated, Inactivate, etc, but I just couldn't get a concise solution without multiple definitions of the same series. I'd like to define target series just once, and then get different symbolic and/or numerical representations of it.

Thank you

---

Bonus question (answered) Is it possible to make Mathematica sum divergent series (Cesàro, Borel, etc) like:

1 - 1 + 1 - 1 ... = 1/2 or 1 - 2 + 4 - 8 ... = 1/3

Update: Regularization parameter of Sum function does the job for divergent series.

Answer 1

With UpValues and Inactive:

noEval /:
 Sum[noEval[expr_],
  (indx : ({_Symbol, _Integer} | {_Symbol, _Integer, _Integer} | {_Symbol, _Integer, _Integer, _Integer}) ..)] :=
   Inactive[Sum][expr, indx] == 
    Inactive[Plus] @@ Flatten@Table[expr, indx] == 
    Sum[expr, indx]

Then

Sum[noEval[n*(-1)^(n - 1)], {n, 1, 10}]

Sum[noEval[i + j], {i, 1, 5}, {j, 1, 2}]

I've just discovered UpValues and am finding questions that suit it quite regularly.

Hope this helps.

Answer 2

Not as sophisticated as the other solutions, but it's good for at least single sums:

Options[displaySum] = {"Terms" -> 10};

displaySum[a_, {n_, n1_, n2_}, opts : OptionsPattern[]] /; n1 <= n2 :=
  Module[{nf = Min[n1 + OptionValue["Terms"] - 1, n2]}, 
         Row[{Defer[Sum[a, {n, n1, n2}]], 
              Composition[Defer, Plus] @@
              Append[Table[a, {n, n1, nf}], If[n2 === ∞, "⋯", Nothing]], 
              Sum[a, {n, n1, n2}]}, "="]]

Some examples:

Answer 3

How about this?:

Clear@seriesExpanded; 
seriesExpanded[expr_, iter : {_, _, _, ___} ..] := 
 Row[If[First@# === "+", Delete[#, 1], #] &@
  Flatten@Replace[
    Flatten@Table[expr, iter], {a : (_?Negative | _?Negative _) :> {"-", -a}, 
     a_ :> {"+", a}}, {1}], " "]
display[a__] := HoldForm@Sum[a] == seriesExpanded[a]

display[(-1)^(i + j) Subscript[a, i, j] j/i, {i, 1, 2}, {j, 2, 5}]