a sequence An is convergent if and only if |An| is convergent. Is it true of false? please give some examples. .....it seems true to me.

Question

a sequence An is convergent if and only if |An| is convergent.  Is it true of false? please give some examples.      .....it seems true to me.

zarkon · Answer

$$A_n=(-1)^n$$

zarkon · Answer

if you start st n=1 then 

$$\{A_n\}=\{-1,1,-1,1,\ldots\}$$

$$\{|A_n|\}=\{1,1,1,1,\ldots\}$$

zarkon · Answer

* start at ...

anonymous · Answer

{An}  and |{An}|  are not convergent in your case.

zarkon · Answer

no..that is incorrect

anonymous · Answer

what do you mean? so can i say {An} is convergent if and only if |{An}| is convergent then?

zarkon · Answer

no

zarkon · Answer

the first sequence i wrote is not convergent...the second is (it converges to 1)

anonymous · Answer

I agree that the first sequence {An} is not convergent.  but i don't get the 2nd one. if the 2nd sequence |{An}| converges to 1, then we have | |{An}| - 1 | < e ?   ...so that we can say |{An}| is convergent?  is tat right?

zarkon · Answer

the second sequence consists for just 1's ...so $$\lim_{n\to\infty}A_{n}=1$$

zarkon · Answer

*consists of just ....

anonymous · Answer

thanks so much for your help!! but.... if the sequence itself is all the 1s, then how could its limit be 1? sorry, i still don't get it.

anonymous · Answer

I think I got it!! Thanks so much!!!

anonymous · Answer

can i ask you one more question: if lim An^2  = 0, then lim An = 0.  is that true or false?  Appreciate it.