I am interested in knowing if this is true or if not or if "it does depend".Say I have a series $$\sum_{k = 0}^{+\infty} a_k\cdot b_k$$
Where $a_k, b_k$ are well defined sequences. Suppose $b_k$ is a geometric sequence (I mean that $\sum b_k$ would constitute a geometric series, which already implies $|b_k| < 1$).
Now suppose $a_k$ is bounded from above, I would ask you if those two cases are formally correct or if there is something more I should say, add, know.
Case $1$: suppose $a_k$ is bounded from above by it's max, reached (say) at $k = k_0$.Suppose I take a set $S$ and $k_0 \not \in S$, but $k_1 > k_0 \in S$. In terms of bounding, I would know if I can write this:
$$\sum_{k = 0}^{+\infty} a_k\cdot b_k \leq \max_S \{a_k\} \sum_{k = k_1}^{+\infty} b_k$$
Pay attention here: the focus of my question is both on the max part, but most of all in the index of the geometric series. Does it start from $k_1$, or am I thinking wrong? Should it start from $k =0$ anyway? Of should it start from $k = k_0$?
Case $2$: $a_k$ is bounded from above but by it's sup. This means $\lim_{k\to +\infty}a_k = M \in\mathbb{R}$. What now? Can I still write something like
$$\sum_{k = 0}^{+\infty} a_k\cdot b_k \leq \sup(a_k) \sum_{k = 0}^{+\infty} b_k$$
or should I make the geometric series start from a different value?
I am confused on how to properly understand those bounding... Thank you!