Why series expansion doesn’t work

Question

The Euler Maclaurin Formula is $$\sum_{k=0}^x f(k)=\int_0^xf(t)dt+\frac{f(x)+f(0)}2+R_1\tag{1}$$Where $R_1$ is the remainder term defined as (according to Wikipedia)$$R_p=\frac{(-1)^{p+1}}{p!}\int_0^xf^{(p)}(t)P_p(t)dt\tag{2}$$I also know that $$\sum_{k=0}^xf(k)=\sum_{k=0}^\infty(f(k)-f(k+x))$$Where $f(k)\rightarrow0$. Taking the derivative on both sides of equations $(1)$ and $(2)$ and setting them equal, we get $$-\sum_{k=0}^\infty f’(k+x)=f(x)+f'(x)\left(P_1(x)+\frac12\right)$$Solving for $f(x)$: $$f(x)=-\left(\sum_{k=0}^\infty f'(k+x)+f'(x)\left(x-\lfloor x\rfloor\right)\right)$$Since $P_1(x)=x-\lfloor x\rfloor -\frac12$. If this series expansion was true, then it would be very well known as we take only the first derivative of the function $f(x)$. This was already a red flag, so I tried $f(x)=\frac1{x+1}$ and I found that it wasn't true. Is there something wrong with my derivation? I think this might have to do with the remainder term (the solution is very straightforward, so making mistakes is unlikely) being false because it comes from Wikipedia. However, I found a PDF from MIT that suggests the same thing written in a different form.