# How to evaluate user-defined functions with variables?

I have an extremely simple question, but I couldn't find anything anywhere, maybe I didn't know what to search for, so feel free to mark it as duplicate if there is a duplicate.

Let's say that I define some function, for example

f[p_Integer] := Sum[k!/(k^2 - 3), {k, 0, p}].

If I want to evaluate it for some integer, for example f[10], it works, as expected. But what if I want to evaluate it for a variable, for example f[n], how do I tell Mathematica to do it? I get f[n] as the result, but that's not so useful.

I'm interested in the general solution, nevermind my example, I had to use something.

Well, you told M that the argument to f must have Integer head. So when you said f[n] and n is not integer, then M did not know what to do and returned back the input. If you wanted f to be evaluated for non integer heads, you can do

ClearAll[n,f,p]
f[p_Integer]:=Sum[k!/(k^2-3),{k,0,p}]
f[p_]:=Sum[k!/(k^2-3),{k,0,p}]  (*or can also use f[p_Symbol]*)


Now

f[5]


f[n]


• It is useful to note that f[5] == (f[n] /. n -> 5) evaluates to True, i.e., the DifferenceRoot gives the same result as the integer input. Consequently, while the definition for integer input is efficient, it is computationally redundant. – Bob Hanlon Nov 27 '17 at 0:25
• Great, thank you, I really needed the f[p_Integer] specification, so I didn't know how to do both, this works! – lel Nov 27 '17 at 1:14