Prove that if the functional sequence ‎$xf(n) + \sum_{k=1}^n f(k)$ ‎is ‎convergent‎ ‎then the sequence ‎$‎f(n)$ ‎is ‎convergent Let ‎$‎f:D_f‎\rightarrow‎\mathbb{R}$ be a function such that $‎‎\mathbb{N}‎‎\subseteq D_f‎$. My question is proving or disprving the following\‎
"If the functional sequence ‎$xf(n) + \sum_{k=1}^n f(k)$ ‎is ‎convergent as ‎$‎‎n\to{‎\infty}‎$‎ ‎‎‎then the sequence ‎$‎‎f(n)$ ‎is ‎convergent." ‎(‎$‎‎x\in‎\mathbb{R}‎$‎)‎\
Note that, if the series ‎$‎\sum_{n=1}^{‎\infty‎} f(n)$ is convergent then getting the result is clear. So, we should find the other cases.
‎

If $g_n(x):=xf(n)+\sum_{k=1}^{n} f(n)$ then:
$$2g_n(x) - g_n(2x) = \sum_{k=1}^{n} f(k)$$
