question about decomposition of positive terms series $\sum_{n=1}^\infty a_k$ where $a_k=b_k + c_k$

If $$\sum_{n=1}^\infty b_n\tag{1}$$ and $$\sum_{n=1}^\infty c_n\tag{2}$$ converge, then it is very easy to show that $$\sum_{n=1}^\infty(b_n + c_n)\tag{3}$$ converges.

The converse is not always true, as the example $b_n=-n, c_n=n$ shows.

The converse can be true under some additional conditions though. For example, if $b_n$ and $c_n$ are all positive, then we can prove that if the sum $(3)$ converges, then the sum $(1)$ also converges.

You can prove this easily because you have $|b_n| = b_n \leq b_n + c_n$. Knowing this, the comparison test immediately gives the result that $(1)$ is a convergent sum.

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