# Convergence/Divergence Proof

1. Mar 3, 2014

1. The problem statement, all variables and given/known data

Show if an > 0 for each n$\in$$N$ and if ∑an converges, then ∑an2 converges and that ∑1/an diverges.

NB. all ∑ are between n=1 and ∞

2. Relevant equations

3. The attempt at a solution

Let partial sums of ∑an2 be Sk = a1 + ... + ak

To say ∑an2 is absolutely convergent is to say ∑|an2| is convergent.

It follows that partial sums Tk = |a12| + |a22| + ... + |ak2| of the series are bounded above by M.

Then by an extended form of the Triangle Inequality we have:

|Sk|

= a12 + a22 + ... + ak2

≤ |a12| + |a22| + ... + |ak2|

= Tk

≤ M

Hence the sequence {|Sk2} is bounded above by M. It is bounded below by 0 as an > 0 so an2 > 0. Therefore it is bounded.

It is therefore convergent.

Is this correct so far? Would a similar proof follow for showing ∑1/an is divergent?

Can anyone help with this please?

2. Mar 3, 2014

### PeroK

I think you're missing a simple solution. Think of what happens to a_n as n gets large.

I don't follow your proof. (How are you using the convergence of the series a_n?)

3. Mar 3, 2014

### micromass

Staff Emeritus
What is $M$ and why does it bound the $T_k$?
You have only established that $\sum a_n^2$ being absolutely convergent is the same as saying that $\sum|a_n|^2$ converges. But you have not established the fact that $\sum a_n^2$ actually absolutely converges!