Consider $(a_n)_n$ and $(w_n)_n$ two sequences of non-negative numbers, with $w_n \geq 1 \forall n $ and $w_n \to \infty$ as $n \to \infty$. I'd like to know what we must assume about the sequences such that the following is true:
There's a constant $C$ such that, for all $N>0$ $$\sum_{n=1}^N a_n w_n \leq C \sum_{n=1}^N a_n$$
Or, in other words,
$$\limsup_{N \to \infty} \frac{\sum_{n=1}^N a_n w_n}{\sum_{n=1}^N a_n}<\infty.$$
Notice that, as $w_n \geq 1$, the sequence $\frac{\sum_{n=1}^N a_n w_n}{\sum_{n=1}^N a_n}$ is increasing, and therefore if the $\limsup$ is finite, then the limit exists.
This is true if both $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty a_n w_n$ are convergent series.
For example take $a_n = \frac{1}{n^3}$ and $w_n = n$.
If however $\sum_{n=1}^\infty a_nw_n=\infty$ but $\sum_{n=1}^\infty a_n<\infty$, then the result is false.
For example take $a_n = \frac{1}{n^2}$ and $w_n = n$.
But what may happen if both series are divergent? I saw in this post https://math.stackexchange.com/questions/92945/is-there-a-discrete-version-of-de-lhôpitals-rule this Stolz-Cesàro theorem (I didn't know previously), stating that
If $b_n$ and $c_n$ are both increasing and unbounded, then $$\lim \frac{b_n}{c_n} = \lim \frac{b_{n+1}-b_n}{c_{n+1}-c_n}$$
and in that case, making $b_n = \sum_{j=1}^n a_jw_j$ and $c_n = \sum_{j=1}^n a_j$, then
$$\lim_{n \to \infty} \frac{b_n}{c_n} = \lim_{n \to \infty} w_{n+1} = \infty.$$
So the only instance in which such sums are comparable is if both of them are convergent? Is this correct?
Thanks in advance.
