I am confused about these $2$ definitions of the Euler-Maclaurin formula.
I read the following here:
The full Euler-Maclaurin formula with no remainder term (for infinitely differentiable $f$ ) is given in Concrete Mathematics [2, p. 471]:
$\sum_{j=0}^{m-1}{f(j)}=\int_{0}^{m}{f(x)dx}+\sum_{k=1}^{\infty}{\frac{B_k}{k!}\left(f^{(k-1)}(m)-f^{(k-1)}(0)\right)}$.
But the wikipedia article found here says this:
Explicitly, for $p$ a positive integer and a function $f(x)$ that is $p$ times continuously differentiable in the interval $[m,n]$, we have
$S-I=\sum_{k=1}^{p}{\frac{B_k}{k!}\left(f^{(k-1)}(n)-f^{(k-1)}(m)\right)}+R_p$
My question is, if a function is infinitely differentiable over a given interval, how can the wikipedia definition hold since infinity is not an integer?
I could not find a reference that gives the definition the way Wikipedia does. Can anyone explain the discrepancy between the two definitions? What am I misunderstanding?