Indeterminate expression. Where?

Question

Consider the following:

a[m_, n_, x_] := 
    1/2^n (-1)^n/((2 n)!) Sum[Binomial[m, l] (m - 2 l)^(2 n), {l, 0, m}] x^(2 n)

a[m_, 0, x_] := 1

f[m_, x_] := Sum[a[m, n, x], {n, 0, Infinity}]

I'm totally clueless why Mathematica fills in $0$ for $x$ while I feed it $x = 1$!

Answer 1

Since you define f[m_,x_] by Sum[a[m,n,x],{n,0,Infinity}] so there are terms in the sum like 0^0 because :

a[0,n,1]=1/2^n ((-1)^n)/((2n)!) Sum[Binomial[0,l] (0-2l)^(2n),{l,0,0}] x^(2n)

When there is a[0,n,1] you start the sum with Sum[Binomial[0,l] (0-2l)^(2n),{l,0,0}] and the first term in the definition of f[m_,x_] is with n==0.

Edit

To prevent generating of messages one can define a[m,n,x] and f[m,x] this way :

a[m_Integer, n_Integer, x_] /; n > 0 := 
  1/2^n (-1)^n/((2 n)!) Sum[ Binomial[m, l] (m - 2l)^(2n), {l, 0, m}] x^(2 n)
a[m_Integer, 0, x_] := 1
f[m_Integer, x_] := Sum[ a[ m, n, x], {n, 0, Infinity}]

Answer 2

It is not because of x, but because the Sum in the definition of a.

In[12]:= Sum[Binomial[m, l] (m - 2 l)^(2 n), {l, 0, m}] /. m -> 0

Out[12]= 0^(2 n)