Finding the Inverse of a Summation I have seen more specific versions of this question but my question is more general.  For any given summation does there exist an inverse.  If not, how does one tell if the function has an inverse.  Do these inverses always have closed forms (I imagine they do not)?  How can one tell when a function such as a summation or its inverse has a closed form and if they do not how would one write them?
This question is motivated by a function I ran across of the form $f(x)=\sum_{a=1}^{x} \sum_{b=1}^{a} b^{b}$ that then required the use of its inverse.  I have thus far failed to write the function or its inverse in closed form.  Thanks for any insight you can provide!
 A: 
For any given summation does there exist an inverse. 

The summation of integers with variable upper limit of the sum produces a map $f:\mathbb N\to\mathbb Z$. In the special case under consideration, where all integers being summed are positive, we have a map  $f:\mathbb N\to\mathbb N$ which is strictly increasing, and therefore injective. As such, it has an inverse $g=f^{-1}$, which is defined on the set $f(\mathbb N)$, the range of $f$. 
Typically, the range of $f$ is not all of $\mathbb N$. For this reason, it is practical to relax the notion of inverse as follows (assuming $f$ is strictly increasing, as above):
$$g(y)=\max\{x:f(x)\le y\} \tag1$$
This $g$ agrees with $f^{-1}$ where the latter is defined. 

How can one tell when a function such as a summation or its inverse has a closed form 

Easily: by drawing from the vast expertise contained in the OEIS and computer algebra systems. There is no realistic hope for explicit form of the inverse function (or the direct function) in your case.

if they do not how would one write them?

Just as   a sum  (or maximum/minimum as in (1)).
