# 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!

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If the innermost sum of your $f(x)$ doesn't even seem to have a closed form, it seems even less likely that $f(x)$ itself has one... – J. M. May 10 '12 at 17:16
I agree that was my initial thought. So how would you write its inverse other then just writing $f^{-1}$ or even prove that it has one? – Jesse Stern May 10 '12 at 17:19
Also, you are going to have a hard time coming up with a closed form for the inverse of $g(x)=x^x$, so already you are dealing with a "difficult" function on the inside of your sum. Not that it helps to get an inverse for $g$ to get an inverse for your $f$, just that these sorts of functions tend to be problematic. – Thomas Andrews May 10 '12 at 17:33