# Thread: Help with proving/disproving statements - series

1. ## Help with proving/disproving statements - series

Hi guys,
i need to prove or disprove these statements:

a. if the series $\displaystyle \sum a_{n}$ is convergent, then $\displaystyle \sum \frac{1}{a_{n}}$ is divergent.

a. if the positive series $\displaystyle \sum a_{n}$ is convergent, then $\displaystyle \sum a_{n}^2$ is convergent.

if the statement is false, i need an example of contradiction.

basically i can really use some tips about how to approach these questions;
what tests should i use to determine if a statment is true or false (or maybe the first step should be merely my intuition), and in what ways can i prove it?

2. ## Re: Help with proving/disproving statements - series

What do you know? Do you know some examples of convergent and divergent series? Do you know "If $\displaystyle \sum a_n$ converges then $\displaystyle lim a_n= 0$"? That should help greatly with the first one. Do you know the "ratio test"? That should help with the second. But be careful, it is precisely the condition under which the ratio test does not apply that is important!

3. ## Re: Help with proving/disproving statements - series

Hi HallsofIvy,

yes, i know that if $\displaystyle \sum a_n$ converges then $\displaystyle lim a_n= 0$ and about the "ratio test", for some unknown reason i didn't think to use it...
thanks a lot!

4. ## Re: Help with proving/disproving statements - series

You can note that if $\displaystyle 0<x<1$ then $\displaystyle 0<x^2<1$.

If $\displaystyle \left( {a_n } \right) \to 0$ then $\displaystyle (\exists N)[n\ge N\to |a_n|<1$.

So use basic comparison.