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