TITLE: Does absolute convergence of a sequence imply convergence?
QUESTION [6 upvotes]: In my real analysis notes I've got that absolute convergence of a real SERIES implies convergence of the series. However what about absolute convergence of a sequence? Does this imply convergence of the sequence?

REPLY [7 votes]: No. e.g. $(-1)^n$ does not converge but $\left|(-1)^n\right|=\left|-1\right|^n=1 \quad $ does converge (to $1$).

As NotNotLogical has pointed out, the exception to this is when a sequence connverges absolutely to $0$, in which case the sequence converges to $0.$