Is there a sequence such that $a_n\to0$, $na_n\to\infty$ and $\left(na_{n+1}-\sum_{k=1}^n a_k\right)$ is convergent? Is there a positive decreasing sequence $(a_n)_{n\ge 0}$ such that
${\it i.}$  $\lim\limits_{n\to\infty} a_n=0$.
${\it ii.}$  $\lim\limits_{n\to\infty} na_n=\infty$.
${\it iii.}$  there is a real $\ell$ such that $\lim\limits_{n\to\infty} \left(na_{n+1}-\sum_{k=1}^n a_k\right)=\ell.$
Of course without condition ${\it i.}$ constant sequences are non-increasing examples that satisfy the other two conditions, but the requirement that the sequence must be decreasing to zero makes it hard to find!.

Assume $\{a_n\}$ exists.
Let $a_n = f (n)$. By i and ii, we have $f (n) = o (1)$ and $f (n) = \Omega (1/n)$. By Euler-McLaurin summation formula, we have $$\sum_{k = 1}^{n} f (k) = c_1 F (n) + o (F (n))$$ for a constant $c_1$, where $F$ is anti-derivative of $f$, and $F (n) = o (n)$ and $F (n) = \Omega (1)$. Then, by iii, we have
$\begin {eqnarray}
n a_{n + 1} - \sum_{k = 1}^{n} a_k & = & n f (n + 1) - \left(c_1 F (n) + o (F (n)\right) \nonumber \\ & = & c_2 F (n) + o (F (n)) - \left(c_1 F (n) + o (F (n))\right) \nonumber \\ & = & (c_1 - c_2) F (n) + o (F (n)) \nonumber \\ & = & O (F (n)) \nonumber \\ & = & \Omega (1),
\end {eqnarray}$
but $\Omega (1)_{n \to \infty} = \infty$, a contradiction. Hence, $\{a_n\}$ does not exist.
