# Math Help - infinite series question

1. ## infinite series question

First of all, I want to say sorry for the language, I thing I speak pretty well English, but mathematics English is an all different game for me...
I'm from Israel BTW.

Ok, now for the questions:
1. I have 2 infinite series, a^2 from 1 to infinite and b^2 from 1 to infinite, both of them converges, I need to prove the infinite series (a * b) from 1 to infinite Absolute convergence

2.
I have the infinite series a^2 from 1 to infinite converges, I need to prove the infinite series (a/n) from 1 to infinite also converges.

I hope you got it, this are the last 2 question I got left in this homework.

2. 1. By Cauchy-Schwarz we have: $
\left( {\sum\limits_{k = 1}^n {a_k ^2 } } \right) \cdot \left( {\sum\limits_{k = 1}^n {b_k ^2 } } \right) \geqslant \left( {\sum\limits_{k = 1}^n {\left| {a_k \cdot b_k } \right|} } \right)^2

$
what do you conclude?

2. Again: $
\left( {\sum\limits_{k = 1}^n {a_k ^2 } } \right) \cdot \left( {\sum\limits_{k = 1}^n {\tfrac{1}
{{k^2 }}} } \right) \geqslant \left( {\sum\limits_{k = 1}^n {\left| {\tfrac{{a_k }}
{k}} \right|} } \right)^2
$

And: $
\sum\limits_{k = 1}^{ + \infty } {\tfrac{1}
{{k^2 }}}
$
converges

3. Originally Posted by asi123
1. I have 2 infinite series, a^2 from 1 to infinite and b^2 from 1 to infinite, both of them converges, I need to prove the infinite series (a * b) from 1 to infinite Absolute convergence

2. I have the infinite series a^2 from 1 to infinite converges, I need to prove the infinite series (a/n) from 1 to infinite also converges.
1)If $\sum_{n=1}^{\infty}a_n^2$ converges and $\sum_{n=1}^{\infty}b_n^2$ converges then $\sum_{n=1}^{\infty} a_n^2b_n^2$ converges but then $\sum_{n=1}^{\infty}\sqrt{a_n^2b^2}$ converges. Now note $\sqrt{a_n^2b_n^2} = |a_nb_n|$.

2) $\left| \frac{a_n}{n} \right| \leq a_n^2 + \frac{1}{n^2}$.

EDIT: Paul was fast.