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!
