Determining convergence of series using "Direct comparison test"

Question

Given that $\sum a_n\leq \sum b_n$, if the series of $b_n$ converges, so does the series of $a_n$.

In my opinion this idea seems to be very general, because it is easy to find $b_n$ bigger than $a_n$ (both have the same domain)

Question: When finding $b_n$, am I bound to certain restriction?

Answer 1

Finding $(b_n)$ sometimes is more tricky than others, but you are right in that it is a fairly general idea. Nevertheless, an extremely useful one.

The only restrictions you have is spelled out in the original quoted statement:

• $\exists N \in \mathbb{N}$ so that $b_n > a_n \ \forall n > N$
• $\sum_{n=N}^\infty b_n < \infty$