Let's break down the problem step by step to understand what is needed to ensure that:

a) $\lim_{n\to \infty} a_n = 0$, or

b) $\sum_{n = 1}^\infty a_n < \infty$

Restating the condition:

We are given the inequality:

$a_n \leq c_n + \sum_{k = 1}^{n-1} a_k \cdot b_{n-k}$

and the fact that $c_n \to 0$ as $n \to \infty$.

This implies that for large $n$, the sequence $c_n$ becomes arbitrarily small, so we are essentially left with the recursive sum:

$a_n \leq \sum_{k = 1}^{n-1} a_k \cdot b_{n-k}.$

Goal (a): $\lim_{n \to \infty} a_n = 0$

To ensure that $a_n \to 0$, we can look for sufficient conditions on the sequence $b_n$ that cause $a_n$ to decrease over time.

• For large $n$, if the sequence $b_n$ decreases sufficiently fast (i.e., $b_n$ converges to 0 rapidly), the recursive sum $\sum_{k = 1}^{n-1} a_k \cdot b_{n-k}$ would be dominated by the terms where $a_k$ is small, resulting in $a_n$ approaching 0 as well.

Sufficient condition for (a):

• A sufficient condition for $a_n \to 0$ is that $b_n \to 0$ sufficiently fast. Specifically, if $b_n$ decays faster than $\frac{1}{n}$, say $b_n = O\left(\frac{1}{n^p}\right)$ for some $p > 1$, then the recursive sum will decay rapidly, leading to $a_n \to 0$.

  More formally, if $\sum_{k=1}^{n-1} a_k \cdot b_{n-k} \to 0$ as $n \to \infty$, then $a_n \to 0$ by the recursive definition.

Goal (b): $\sum_{n = 1}^\infty a_n < \infty$

For the sum $\sum_{n = 1}^\infty a_n$ to converge, we need the sequence $a_n$ to decay fast enough.

• From the recursive relation, $a_n$ is bounded by a weighted sum of previous terms. If the weights $b_{n-k}$ decay fast enough, then the terms $a_n$ should decrease rapidly enough for the series to converge.

Sufficient condition for (b):

• For convergence of the series $\sum_{n = 1}^\infty a_n$, we need the sum $\sum_{n = 1}^\infty b_n$ to converge. This would force the sequence $a_n$ to decay at a rate fast enough that the total sum remains finite. In particular, if $b_n$ is chosen such that $\sum_{n = 1}^\infty b_n$ converges, then the series $\sum_{n = 1}^\infty a_n$ should also converge.

Conclusion

• For (a) $\lim_{n \to \infty} a_n = 0$, a sufficient condition is that $b_n$ decays faster than $\frac{1}{n}$, i.e., $b_n = O\left(\frac{1}{n^p}\right)$ for some $p > 1$.
• For (b) $\sum_{n = 1}^\infty a_n < \infty$, a sufficient condition is that $b_n$ is chosen such that $\sum_{n = 1}^\infty b_n$ converges.

Would you like further details on how to derive these conditions, or any specific examples?