Prove or disprove the existence of f(k) when $\sum_{n=1}^m{\frac{a_n}{n^s}}=\sum_{n=1}^m\frac{a_n+f^n(x)}{n^s}$.

Question

Can we find a function, $f(k)$, such that $$\frac{a_n}{n^s}+\frac{a_{n+1}}{(n+1)^s}+\frac{a_{n+2}}{(n+2)^s}=\frac{a_n+x}{n^s}+\frac{a_{n+1}+f(x)}{(n+1)^s}+\frac{a_{n+2}+f(f(x))}{(n+2)^s}$$ for some $x$?

Could it possibly be shown in the general case, in terms of the iterated function in $$\sum_{n=1}^m{\frac{a_n}{n^s}}=\sum_{n=1}^m\frac{a_n+f^{n-1}(x)}{n^s}$$?