### Abstract:

It is well-known that the sum of integers from 1 to n is . But what
2 happens when we add these sums together? Do we have a closed form formula
ni
nn
for j? What happens if we keep repeating the process? In general, do we
i=1 j=1
na1an−1
nn n
have a closed form formula for ... an? We will then look at the sum of
a1=1 a2=1 an=1 squares, cubes,... of consecutive integers and repeat the process. Would we always get a nice closed form formula for the finite multiple sum of powers of integers? Are there are other sequence that we can compute the partial multiple sum? In this thesis we will discover it all.