1) Suppose $\sum _{n=0}^\infty a_n$ and $\sum _{n=0}^\infty b_n$ both converges.
a) Does $\sum _{n=0}^\infty (a_n+b_n)$ converge always, sometimes or never?
b) Does $\sum _{n=0}^\infty (a_n*b_n)$ converge always, sometimes or never?
c) Does $\sum _{n=0}^\infty max(a_n,b_n)$ converge always, sometimes or never?
d) Does $\sum _{n=0}^\infty min(a_n,b_n)$ converge always, sometimes or never?
2) Suppose $\sum _{n=0}^\infty a_n$ converges and $\sum _{n=0}^\infty b_n$ diverges.
a) Does $\sum _{n=0}^\infty (a_n+b_n)$ converge always, sometimes or never?
b) Does $\sum _{n=0}^\infty (a_n*b_n)$ converge always, sometimes or never?
c) Does $\sum _{n=0}^\infty max(a_n,b_n)$ converge always, sometimes or never?
d) Does $\sum _{n=0}^\infty min(a_n,b_n)$ converge always, sometimes or never?
This is my answer:
For 1) because $\sum _{n=0}^\infty a_n$ and $\sum _{n=0}^\infty b_n$ both converges, $a_n and b_n$ converges to zero, so $a_n+b_n$ converges to zero
a) Since $\sum _{n=0}^\infty (a_n+b_n)=\sum _{n=0}^\infty a_n +\sum _{n=0}^\infty b_n$, so yes it always converges.
b) Since $\sum _{n=0}^\infty (\alpha a_n)= \alpha \sum _{n=0}^\infty a_n $ and since $b_n ->0$ we can treat it as a constant $\alpha$, so yes it always converges.
For c), and d) I'm not sure how to reasoning these.
For 2) because $\sum _{n=0}^\infty a_n$ converges and $\sum _{n=0}^\infty b_n$ diverges , so $a_n ->0$ but $b_n$ may not converge to zero.
a) Since $\sum _{n=0}^\infty (a_n+b_n)=\sum _{n=0}^\infty a_n +\sum _{n=0}^\infty b_n$, so yes it sometime converges.
b) Since $\sum _{n=0}^\infty (\alpha a_n)= \alpha \sum _{n=0}^\infty a_n $ and since $b_n$ may or may not converge to zero, so we can't treat it as a constant $\alpha$ all the time, so yes it sometime converges.
For c), and d) I'm not sure how to reasoning these.
Is my reasoning correct? I'm working on making convincing argument, so please feel free to point out my mistake.
