Value assignment to a (pseudo) function that is nowhere finite .

Question

Is there a method in mathematics literature which helps us to assign values (equivalent finite function) ( like analytic continuation but not analytic continuation) for (pseudo) functions that are nowhere finite ?

Example:

Can we assign a "finite function" for the given (pseudo) function ? :

$$f(x)= \sum_{n=1}^\infty e^{x/n²}$$