Relation Between Binomial Sums and Sequences

Question

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.

Answer 1

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 $O(\frac{n}{\log(n)})$.