Given 2 series' an bn, what can be said about cn when

[Dell]

given 2 unknown number series {a_n}$$^{infinity}_{n=1}$$ and {b_n}$$^{infinity}_{n=1}$$, if it is know that {a_n}$$^{infinity}_{n=1}$$=infinity and {b_n}$$^{infinity}_{n=1}$$=K,
K=constant unknown

what can be said about
{a_n+b_n}$$^{infinity}_{n=1}$$
{a_n*b_n}$$^{infinity}_{n=1}$$


i think that
{a_n+b_n}$$^{infinity}_{n=1}$$= infinity, since:
{a_n+b_n}$$^{infinity}_{n=1}$$={a_n}$$^{infinity}_{n=1}$$+{b_n}$$^{infinity}_{n=1}$$= infinity + K = infinity

as for {a_n*b_n}$$^{infinity}_{n=1}$$, i don't think anything can be said for sure, since K could be a number or 0,

am i right?? is there a proper mathematical way to write this?

 

[MathematicalPhysicist]

Partially right.
If K=0 it's usually undefined, but if K>0 then K*infinity=infinity and if K<0 then... what can you say it equals to?

If you need to check the definitions there they are:
1. a_n=infinity iff for every M>0 there exists N such that for every n>N a_n>M
2. a_n=-infinity iff for every M<0 there exists N such that for every n>N a_n<M.

 

[Dell]

i am asked is Cn converges when
Cn=An+Bn,---> does not converge

Cn=AnBn--->if |Bn|>0 - does not converge
but if Bn=0, then it is undefined, so what does that mean, diverge/converge/something else??

 

[MathematicalPhysicist]

Well, now when I think of it if Bn=0, and An->infinity then AnBn->0 because AnBn=0 for every natural n.
If the question were, Bn->0 but not indetically 0, then it's undefined.