Prove Convergence of Positive Series Squares

[Reid]

Homework Statement

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!

Homework Equations

The relevant equations has been stated above.

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!

 

[D H]

Apply the ratio test.

 

[Reid]

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

Thank you so much! :)

 

[harryjose]

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

 

[HallsofIvy]

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

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

Please, please, please do not "hijack" someone else's thread to ask your own question! It is very easy to start your own thread.

 

[Reid]

I can only speak for myself and I can't see the connection to my thread!