Lets imagine, that we have a random sequence, which is convergent towards 0. And another sequence that is limited. Now how would I show that these two sequences multiplied together is convergent?

Question

anonymous · Answer

Would it make sense to use Cauchy sequence?

astrophysics · Answer

Maybe did you try it

astrophysics · Answer

What does it mean by convergent towards 0

anonymous · Answer

We have that the sequence is convergent towards 0, that means when n goes to infinity the sequence approaches 0.

astrophysics · Answer

Yeah I figured as much

anonymous · Answer

So since the other sequence is limited, it makes sense that if you multiply the limits (0*a) it is 0, and therefore convergent towards 0 aswell. But I dont know how to "prove" it.

astrophysics · Answer

Isn't it true if the limit exists and is finite the series is convergent

astrophysics · Answer

Yeah that sounds right

astrophysics · Answer

maybe try partial sums

anonymous · Answer

I need to prove that the 2 sequences multiplied together is convergent. So I believe that I have to do this with the convergent definiton or Cauchy sequence-

astrophysics · Answer

That could work, @ganeshie8 is pretty good with that method, but he's offline right now...a lot of people are at this time haha.

astrophysics · Answer

OH! @mukushla :)

anonymous · Answer

Hey Astro :)

astrophysics · Answer

Hey :D
Maybe you can help with this problem :)

anonymous · Answer

This may help: http://math.stackexchange.com/questions/202467/cauchy-product-of-two-absolutely-convergent-series-is-absolutely-convergent-ru

anonymous · Answer

But there, they are given 2 sequences which both is convergent.

astrophysics · Answer

But if a sequence is limited isn't it also convergent...divergent would be if it does not exist or goes to positive/ negative infinity

anonymous · Answer

What if we have a sequence: $$a_n=i^n$$
This sequence would be limited. but not convergent?

anonymous · Answer

It has to be limited and monotonic, in order to be convergent.

astrophysics · Answer

Ah, yeah I do remember something like that but eh, I don't want to say anything right now as I may just lead you in the wrong direction. I think you should wait for someone who has an expertise in this.
@freckles @ganeshie8 @SithsAndGiggles

astrophysics · Answer

Thought muku left haha xD

anonymous · Answer

Oh wait, sry, we're talking about sequences not series, the link I've provided would not  help.

anonymous · Answer

attachment

anonymous · Answer

It is with convergent definition :)

anonymous · Answer

Thank you, that is a great help :) - I'll take a look at it