Absolute Value Theorem for Sequence

Question

Given an alternating sequence.

The Absolute Value Theorem states that: If the limit of the absolute value of the sequence is 0 then the limit of the original sequence is also 0.

However if, the limit of the absolute value of the sequence is not 0 (some value) then there is "no conclusion".

But wouldn't that just imply that the sequence would alternate between this value making the limit DNE and therefore the sequence divergent? Or is this only for all cases where the sequence is in fact alternating.

Answer 1

The resulting limit from using the Absolute Value Theorem must be 0 to show convergence of the original sequence.

The resulting limit can also be a non zero number which, in general means that there is no conclusion.

However, when the resulting limit is a non zero number and the original sequence is an alternating sequence, you can conclude that the sequence will alternate between this non zero number. Making the limit of the original sequence DNE which implies that the sequence is divergent.

Answer 2

The continuity of $x \mapsto |x|$ implies that the convergence of $\langle a_j \rangle$ implies the convergence of $\langle |a_j| \rangle$, so taking the contrapositive gives the divergence of $\langle a_j \rangle$.

Answer 3

Assume that $$\lim_{n\to+\infty}|x_n |=L>0$$

if, for enough great values of $n $, $x_n$ keeps a constant sign then the sequence $(x_n) $ will converge to $L $ or $-L $.

if its sign changes, it will diverge.