Can the Modulus Rule Be Reversed?

[Silversonic]

Homework Statement

Give an example to show that the modulus rule cannot be reversed. Hence give an example of a divergent sequence ([itex]a_{n}[/itex]) such that (|[itex]a_{n}[/itex]|) is convergent.

Homework Equations

The modulus rule is;

"Let [itex]a_{n}[/itex] be a convergent sequence.

(|[itex]a_{n}[/itex]|) is convergent, then

lim|[itex]a_{n}[/itex]| = |lim[itex]a_{n}[/itex]|"

n is an element of the natural numbers of course, and the limit is the limit as n tends to infinity.

The Attempt at a Solution

I don't understand what it means by "reverse" of this rule. I assumed originally that it meant give an example of a divergent sequence that wouldn't work, but the "hence" bit afterwards would suggest I have to do the same thing twice, which I'm guessing isn't right. I also have another "show the reverse doesn't work for this rule" question before it, but I'm not entirely sure what is meant by reverse.

 

[HallsofIvy]

The reverse (more technically the "converse") would be "If [itex]|a_n|[/itex] converges, then [itex]a_n[/itex] converges and [itex]lim |a_n|= | lim a_n|[/itex]". To show that is NOT true, find a sequence that does NOT converge but the absolute value does not converge.

 

[Silversonic]

The reverse (more technically the "converse") would be "If [itex]|a_n|[/itex] converges, then [itex]a_n[/itex] converges and [itex]lim |a_n|= | lim a_n|[/itex]". To show that is NOT true, find a sequence that does NOT converge but the absolute value does not converge.

Thanks for the reply. But is the bolded bit right? The "but" you put suggested to me you meant to say "does" as opposed to "does not". If that's true, then e.g. [itex](-1)^n[/itex] would be my example. I'm just confused as to why my example sheet says "Hence" show there's is a divergent sequence ([itex]a_n[/itex]) such that |[itex]a_n[/itex]| is convergent, I assume he meant "in other words" as opposed to "hence"?

 

[HallsofIvy]

Yes, that was a typo. It should have been "does converge". And, I think you have a good point about "in other words" rather than "hence".