Let $\sum\limits_{n=0}^{\infty}a_n$ and $\sum\limits_{n=0}^{\infty}b_n$ be convergent with $a_n,b_n\geq 0$, does $\sum\limits_{n=0}^{\infty}\min\{a_n,b_n\}$ and $\sum\limits_{n=0}^{\infty}\max\{a_n,b_n\}$ converge too?
I know, that this was asked here and here in a kind of similar way. The thing is, that in the first link the answer is quite undetailed and the second link contains a different question that is similar but not the same. (That's why I will ask the question yet again.)
My thoughts are, that if both series $\sum_{n=0}^{\infty}a_n,\sum_{n=0}^{\infty}b_n$ converge, that $\sum_{n=0}^{\infty}\min\{a_n,b_n\}$ will pick either value of one of the partial sums and likewise, $\sum_{n=0}^{\infty}\max\{a_n,b_n\}$ will pick a partial sum out of both. With the help of the definition of series, "The series $\sum_{n=0}^{\infty}a_n$ converges, if the partial sum $\sum_{k=0}^{\infty}a_k$ with $k\geq n$ converges", can we apply, that both $\sum_{n=0}^{\infty}\min\{a_n,b_n\}$ and $\sum_{n=0}^{\infty}\max\{a_n,b_n\}$ converge.
Is this enough for a right proof?
Best Answer
Since I don't know what “will pick either value of one of the partial sums” means, I can't tell whether you are right or wrong. But you can do it as follows: since both series $\sum_{n=0}^\infty a_n$ and $\sum_{n=0}^\infty b_n$ converge, the series $\sum_{n=0}^\infty(a_n+b_n)$ converges too. And since$$(\forall n\in\mathbb Z_+):\min\{a_n,b_n\},\max\{a_n,b_n\}\leqslant a_n+b_n,$$both series $\sum_{n=0}^\infty\min\{a_n,b_n\}$ and $\sum_{n=0}^\infty\max\{a_n,b_n\}$ converge, by the comparison test.