Does absolute convergence of a sequence imply convergence?

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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?

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1 Answer

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No. e.g. $(-1)^n$ does not converge but $\left|(-1)^n\right|=\left|-1\right|^n=1 \quad $ does converge (to $1$).

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

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