Replace a family of coefficients by a function in a sum

Question

I would like to do something very simple, i.e. replace the coefficients in a sum by a function:

c[1] + c[2] + c[3] /. {c[s] -> (s + 1)/(2 (2 s + 1))}

Unfortunately, this does not work. Any suggestion for something handy that would do the job?

Answer 1

I think you need to use RuleDelayed like this:

c[1] + c[2] + c[3] /. {c[s_] :> (s + 1)/(2 (2 s + 1))}
(* 193/210 *)

This replaces every coefficient with a formula and evaluates for the integer values.