If {$|a_n|$} is divergent, can {$a_n$} be convergent? If {$a_n^2$} is convergent, can {$a_n$} be divergent?

Question

Can anyone help me prove these problems?

If {$|a_n|$} is a divergent sequence, can {$a_n$} be a convergent sequence?

If {$a_n^2$} is a convergent sequence, can {$a_n$} be a divergent sequence?

Answer 1

For the first part, try to prove by contradiction. That is, assume that $\{a_n\}$ converges to some limit $a$ and show that $\{|a_n|\}$ converges. What will the limit of the latter sequence be?

For the second part, try to construct a sequence that repeatedly "hops" between two values, say $-1$ and $1$.

Answer 2

If {|$a_n$|} is a divergent sequence, can {$a_n$} be a convergent sequence?

This is false. As by a theorem: if the sequence {$a_n$} converges to A, then {|$a_n$|} converges to |A|.

The contrapositive states: If {|$a_n$|} is divergent then {$a_n$} is divergent.

If {$a_n^2$} is a convergent sequence, could {$a_n$} be a divergent sequence?

Consider $a_n=(-1)^n$

the square converges while the actual sequence does not