How to Sum with indefinite variables?

Question

I want something like

f[n] = Sum[2^Sum[i_k, {k, n}], {i_1, Infinity}, {i_2, Infinity},...{i_n, Infinity}]

I'm playing with:

Table[i_k, {k, n}]

But can't find a way to parse it in the definition.

What's the good way to do it?

Answer 1

Try this:

With[{n = 5}, 
     Inactive[Sum][a^Sum[K[i], {i, n}], ##] & @@ 
     Table[{K[i], 1, ∞}, {i, n}]]

$$\sum_{\mathtt{K[1]}=1}^\infty \sum_{\mathtt{K[2]}=1}^\infty \sum_{\mathtt{K[3]}=1}^\infty \sum_{\mathtt{K[4]}=1}^\infty \sum_{\mathtt{K[5]}=1}^\infty a^{\mathtt{K[1]}+\mathtt{K[2]}+\mathtt{K[3]}+\mathtt{K[4]}+\mathtt{K[5]}}$$

Replace Inactive[Sum] with Sum to see the actual result of the evaluation.