Let $\{a\_k\}(k\ge 0)$ be a sequence of nonzero real numbers which changes signs infinitely often. Suppose $|a\_k|\to 0 $ and $|a\_k|$ decreases fast. Let $n$ be a positive integer. What's the relation between
$$\sum\_{k\ge 0}\binom {n+k}{k}a\_k $$ and $$\sum\_{k\ge 0}\binom {n+k}{k}\frac{a\_k}{k}. $$ For the asymptotic behavior, as $n\to \infty$, are $$\frac{\sum\_{k\ge 0}\binom {n+k}{k}a\_k}{\sum\_{k\ge 0}\binom {n+k}{k}\frac{a\_k}{k}}=O(1)$$ and
$$\frac{\sum\_{k\ge 0}\binom {n+k}{k}a\_k}{\sum\_{k\ge 0}\binom {n+k}{k}\frac{a\_k}{k}}=O(\frac{n}{\log n})$$ true or not? I tried to use Abel's summation formula, but it seems that Abel's summation formula is not applicable and I don't find relevant references.



Relation between $\sum_{k\ge 0}\binom {n+k}{k}a_k $ and $\sum_{k\ge 0}\binom {n+k}{k}\frac{a_k}{k}$
Since the second sum cannot start at $k=0$, I assume that both sums start at $k=1$.


Consider a particular example: $a\_k = k\alpha^k$ with $|\alpha|<1$. Then
$$\sum\_{k\geq 1} \binom{n+k}k \frac{a\_k}k = (1-\alpha)^{-(n+1)}-1$$
and
$$\sum\_{k\geq 1} \binom{n+k}k a\_k = (n+1)(1-\alpha)^{-(n+2)}\alpha.$$
Then the ratio of the two $\approx \frac{(n+1)\alpha}{1-\alpha}$, which is not $O(\frac{n}{\log(n)})$.


