Sum of series scaled by the harmonic series

Question

Let $\{ a_k \}$ so that $\sum_{k=1}^n a_k = A_n$. We can also assume that $A_n \rightarrow A$.

It is known that $\sum_{k=1}^n \frac1k \approx \ln(n)$. Is there a way to approximate $\sum_{k=1}^n \frac{a_k}{k}$ ?

Answer 1

If we suppose that the sequence $(a_k)_{k>0}$ is positive, then $(A_k)_{k>0}$ is increasing, so by Abel summation we get : $$\sum_{k=1}^n\frac{a_k}{k}=\frac{A_n}{n+1}+\sum_{k=1}^nA_k\biggl(\frac{1}{k}-\frac{1}{k+1}\biggr)$$ Therefore we have this inequality : $$\sum_{k=1}^n\frac{a_k}{k}⩽\frac{A}{n+1}+A\sum_{k=1}^n\frac{1}{k}-\frac{1}{k+1}=A$$ Or using Landau notation, $$\sum_{k=1}^n\frac{a_k}{k}=O(1)$$