Extracting order n coefficients from a non-closed form summation

Question

I am trying to evaluate the following expression when $c=0$:

$$\sum _{k=1}^{\infty} \dfrac{(k+1) c^{k-1} g(k)}{(k-1)!}+\left(\sum_{k=0}^{\infty} \dfrac{(k+1) c^k g(k)}{k!}\right)^2$$

(copyable Mathematica plaintext version):

Sum[((k+1) c^(k-1) g[k])/(k-1)!, {k, 1, Infinity}] + 
(Sum[((k+1) c^k g[k])/k!, {k, 0, Infinity}])^2

but when I tell Mathematica to do so, it simply gives me:

$$\sum_{k=1}^{\infty} \dfrac{0^{k-1} (k+1) g(k)}{(k-1)!}+\left(\sum _{k=0}^{\infty} \dfrac{0^k (k+1) g(k)}{k!}\right)^2$$

So to find the result that I am looking for, I suppose I should somehow isolate the order zero terms $\cal{O}(c^0)$ and then evaluate these when $c=0$, so the final result is:

$$2g(1)+g(0)^2$$

Is there any way of obtaining the $0$th order coefficients from a non-closed expression as the one I have using only Mathematica?

Answer 1

First, you should always include copyable Mathematica plaintext, so that people can help you without having to type in your input. I modified your question to include a copyable version.

Now, one idea is to replace c^n_ instead of c.

expr = Sum[((k+1)c^(k-1)g[k])/(k-1)!,{k,1,Infinity}] + 
(Sum[((k+1)c^k g[k])/k!,{k,0,Infinity}])^2;

expr /. c^n_ :> Piecewise[{{1, n==0}}, 0]

g[0]^2 + 2 g[1]

This won't work if the exponent can be negative, although in that case you can do something like:

Sum[c^k g[k], {k, -1, Infinity}] /. c^n_ :> Piecewise[{{Infinity, n<0}, {1, n==0}}, 0] //InputForm

Sum[g[k]*Piecewise[{{Infinity, k < 0}, {1, k == 0}}, 0], {k, -1, Infinity}]

and at least Mathematica will indicate that there is an issue.

Answer 2

Maybe these lines can help you to find the appropriate command for your case:

(* 1 *)    
c^k /. c -> 0

(* Out[314]= 0^k *)

(* 2 *)
Assuming[{k >= 0}, c^k /. c -> 0]

(* Out[321]= 0^k *)

(* 3 *)
Limit[c^k, c -> 0, Assumptions -> k >= 0]

(* Out[320]= 0 *)

(* 4 *)
Limit[c^k, c -> 0, Assumptions -> k == 0]

(* Out[319]= 1 *)