Prevent a sum from simplifying?

Question

Basically, I want to prevent a sum from cancelling or summing together the individual summands and just display them separately next to each other. Example:

Sum[ii,{ii,1,3}]

1+2+3

Mathematica either gives back the result 6, or does not perform the sum at all if I use diverse Hold functions:

Sum[ii,{ii,1,3}]//Hold

Hold[Sum[ii,{ii,1,3}]]

Is there any way to get the desired "halfway evaluated" output? I'd like to see every summand separately to make sure that the computation works correctly when it does a very complicated sum of variables but then reduces (mostly cancels) to a very simple answer. Thanks for any suggestion!

Answer 1

Simple: use Table

Sum[i^2, {i, 3}] (* 14 *)

Table[i, {i, 3}] (* {1, 4, 9} *)

If you want the output to be displayed like a sum, use Row:

Row[%, "+"] (* 1 + 4 + 9 *)

And when you want to evaluate, simply use Total:

Total[%%] (* 14 *)

There is another solution (see David's comment):

Sum[(HoldForm[#] &)[i^2], {i, 3}] (* 1 + 4 + 9 *)

To explain how this works, let's look at it piece by piece: First, we create a pure function using & and #. Here are a couple of (mostly) equivalent definitions to explain it:

f[x_] := x^2 - x + 1
f = Function[x, x^2 - x + 1];
f = Function[#^2 - # + 1];
f = (#^2 - # + 1)&;

You can see that in this particular case the function simply applies HoldForm to the argument. We apply this function (with the regular [] syntax) to each summand. This has the effect of wrapping each summand in HoldForm before adding them together, preventing further evaluation of the function while displaying them normally:

%//InputForm (* HoldForm[1] + HoldForm[4] + HoldForm[9] *)

The trick is that simply using HoldForm[i] won't work, since HoldForm will prevent the summand from being evaluated, and it will remain i^2: you'd get i^2 + i^2 + i^2. However, the argument to our anonymous function is evaluated before being passed in, so HoldForm is applied after the summand is evaluated.

To evaluate, simply call ReleaseHold on the expression:

ReleaseHold[%] (* 14 *)

This removes all the HoldForms from the expression, allowing it to be evaluated completely.