# Infinite series

1. Jan 15, 2008

### Reid

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

The infinite series defined by $$\Sigma a_{n}$$, with $$a_{n}>0$$ are convergent. If then the series defined by $$\Sigma a_{n}^{2}$$ coverges, prove it!

2. Relevant equations

The relevant equations has been stated above.

3. The attempt at a solution

Since every term in the first infinite series are positive the partial sums are monotone increasing. And, since it converges these will be bounded above. Then it feels like the series of the squares will be bounded above as well. Since, due to convergence, every term approaches zero.

Is it correct to say that since the term $$a_{n}$$ tends to zero as n tends to infinity, its square also will?

Are my reasoning correct? How am I supposed to do it formally?

So very grateful for hints!

2. Jan 15, 2008

### D H

Staff Emeritus
Apply the ratio test.

3. Jan 15, 2008

### Reid

Ohh... I was making it harder than it actually was!

Thank you so much! :)

4. Jan 17, 2008

### harryjose

find out the sum of arithematic series which has 25 terms and its middle number is 20

5. Jan 17, 2008

### HallsofIvy

Staff Emeritus
Does this have anything at all to do with the original question?