1. convergence

so confused!
if sum (a_n) with a_n > 0 is convergent then is sum (a_n)^2 always convergent? how do we prove it or what is a counter example for it

2. Simple counterexample: the sum $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^{2}}$ converges, the sum $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}$ diverges...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. thankyou

hi thanks for your help but then what is the answer to
if sum (a_n) is convergent then sum (a_n)^2 that is sum of a_n squared is always convergent? counter example or proving it

4. Originally Posted by sandy
hi thanks for your help but then what is the answer to
if sum (a_n) is convergent then sum (a_n)^2 that is sum of a_n squared is always convergent? counter example or proving it
I think chisigma got the question wrong way around.

Think about it this way, if $\displaystyle \sum a_n < \infty$, then $\displaystyle \left (\sum a_n \right)^2 < \infty$.

What does that tell you about $\displaystyle \sum a_n^2$?

Hint: $\displaystyle x^2+y^2 \leq x^2+y^2+2xy=(x+y)^2$ for x,y positive.

5. Originally Posted by sandy
hi thanks for your help but then what is the answer to
if sum (a_n) is convergent then sum (a_n)^2 that is sum of a_n squared is always convergent? counter example or proving it
That is true only if $\displaystyle \forall n$ $\displaystyle a_{n}>0$ or $\displaystyle \forall n$ $\displaystyle a_{n}<0$. An example: $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}}$ converges, $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n}$ diverges...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. $\displaystyle \sum\limits_n {\frac{{( - 1)^n }}{{\sqrt n }}}$ converges.
Does $\displaystyle \sum\limits_n {\frac{{1 }}{{ n }}} ?$