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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?

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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 $b_n > a_n \ \forall n > N$
  • $\sum_{n=N}^\infty b_n < \infty$
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