Iterated sums and asymptotics

Question

Let $x$ be a positive integer. I have the following iterated sums:

$$f(x)=\sum_{i_{u-1}=1}^{x}\sum_{i_{u-2}=1}^{i_{u-1}} \cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1$$

What is the asymptotic behavior of the total?

Thank you

Answer 1

\begin{eqnarray*} \sum_{i_1=1}^{i_2}i_1 & =\frac{i_2(i_2+1)}{2!} \\ \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1 & =\frac{i_3(i_3+1)(i_3+2)}{3!} \\ \vdots & \sum_{i_u-1=1}^{x}\cdots \sum_{i_2=1}^{i_3} \sum_{i_1=1}^{i_2}i_1 & =\frac{x(x+1) \cdots (x+u-1)}{u!} \end{eqnarray*} So the asymtopic behaviour is $x^u/u!$.

Answer 2

You can un-expand the inside by noticing

$$i_1=\sum_{k=1}^{i_1}1$$

Which gives the combinatorial sum:

$$f_u(x)=\binom{x+u-1}u$$

Think of how many ways you can count...

This then gives the following asymptote:

$$f_u(x)\sim\frac1{u!}x^u$$