I have learnt that if $$\sum_{n=1}^\infty a_n $$ converge to A and $$\sum_{n=1}^\infty b_n $$ converge to B, $$\sum_{n=1}^\infty a_n + b_n $$ converge to A+B.
However, if $$\sum_{n=1}^\infty a_n + b_n $$ converges, does it mean that both $$\sum_{n=1}^\infty a_n $$ and $$\sum_{n=1}^\infty b_n $$ converge as well? Thank you in advance!
Not without additional conditions on the series. For example, if $a_n = 1$ and $b_n = -1$ for all $n$, then $a_n + b_n = 0$ and so $\sum (a_n + b_n) = 0$, but clearly the two individual series do not converge.
If you know that $a_n$ and $a_n + b_n$ both converge, then you can definitely conclude that $b_n$ converges because $b_n = (a_n + b_n) + (-a_n)$ and so you can treat it as a sum of convergent series in disguise.
Alternatively, if you have, for example, $a_n, b_n \geq 0$ and $\sum a_n + b_n$ converges, then you can conclude that $\sum a_n$ and $\sum b_n$ both converge because of the dominated convergence theorem.
