Hi,

There are two possible different questions here: how to perform this type of computation and how to typeset this type of computation. The first question is easy and it is what I'll address here. There is no built-in help to typeset this type of expression, but if this is what you are looking for, ask again and we'll look for something.

I suspect there is something wrong in the given expression. First, the i0 index starts at 1 and ends at 0. Can you check that? Then the i1 index is repeated. Then the ik index ends at Floor[n/n], while I would expect it to end at Floor[n/k], if I'm understanding correctly what the expression is. Assuming I'm correcting appropriately all those issues, you could do something like this:

MultiSum[summand_, is_, multis_] := Sum@@ {summand, Splice@Table[multis, is]}

sum[k_, n_] := Block[{i, a}, 
    MultiSum[
        Sum[5 i[a], {a, 0, k}],
        {a, 0, k},
        {i[a], 0, If[a == 0, 1, Floor[n/a]]}
    ]
]

Then you can compute

In[3]:= sum[0, n]
Out[3]= 5

In[4]:= sum[1, n]
Out[4]= 5/2 Floor[n] (1 + Floor[n]) + 5/2 (1 + Floor[n]) (2 + Floor[n])

In[5]:= sum[2, n]
Out[5]= 5/2 (1 + Floor[n/2]) (1 + Floor[n]) (Floor[n/2] + Floor[n]) + 5/2 (1 + Floor[n/2]) (1 + Floor[n]) (2 + Floor[n/2] + Floor[n])

In[8]:= Table[sum[k, 10], {k, 0, 15}]
Out[8]= {5, 605, 5280, 25080, 83160, 273240, 570240, 1188000, 2471040, 5132160, 10644480, 10644480, 10644480, 10644480, 10644480, 10644480}

This is not a well-posed question.

• Have you read the documentation? You can just type "..." into the help tool and find out how to use "..." (aka RepeatedNull)
• What problem around infiinite sums are you struggling with? We can't read your mind.
• What code have you tried?