# Thread: Series: Comparison test

1. ## Series: Comparison test

Suppose for an >= 0 for all n and \sum_{n=1}^\infty a_n is convergent. explain why an < 1 for all large enough n.
Deduce that \sum_{n=1}^\infty \a_n^\2 is convergent.

Could anyone lend a hand? I know you have to use the comparison test, but how?
Thanks

2. ## Re: Series: Comparison test

Originally Posted by algebra123
Suppose for an >= 0 for all n and \sum_{n=1}^\infty a_n is convergent. explain why an < 1 for all large enough n.
Deduce that \sum_{n=1}^\infty \a_n^\2 is convergent.
First, you must explain why it must be true that the sequence $(a_n)\to 0$.

Once you have that $(\exists N)[n\ge N\text{ implies }|a_n|<0.5]$

Now you know that the geometric series $\sum\limits_{k = N}^\infty {{{\left( {0.5} \right)}^k}}$ converges.

3. ## Re: Series: Comparison test

I don't understand how you would know that (a_n) tends to 0 ?

4. ## Re: Series: Comparison test

Originally Posted by algebra123
I don't understand how you would know that (a_n) tends to 0 ?
Then you have utterly failed to learn the first and perhaps most important theorem dealing with series convergence.

Go look up the first test of divergence: If $\sum\limits_{k = N}^\infty {{a_k}}$ convergences only if what?

5. ## Re: Series: Comparison test

Our lecturer is really far behind and hasn't been through any of this. I'm so confused, where did the 0.5 come from?

6. ## Re: Series: Comparison test

Originally Posted by algebra123
Our lecturer is really far behind and hasn't been through any of this. I'm so confused, where did the 0.5 come from?
I doubt we can help with what your lecturer has or has not done.
That is a matter for you to take up with your educational authority.

But you have a textbook/lecture notes. You can at least look up the what part of my question:
$\sum\limits_{k = N}^\infty {{a_k}}$ convergences only if what?
If you want more help, show some effort.