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Given any series (in the real numbers)

inf
---
\ an
/__
n=1

we can consider the series

inf
---
\ |an|  = |a1| + |a2| +...
/__
n=1

whose terms are the absolute values of the terms of the original series. If this series converges, then the original series is said to be absolutely convergent. (Note that for series with positive terms, absolute convergence is the same as convergence).

The series

inf
---
\ ((-1)n-1)/n
/__
n=1

is convergent (we know this by the alternating series test). However, the corresponding series of absolute values is divergent (it is the harmonic series). Series that converge without being absolutely convergent are called conditionally convergent.

All series that are absolutely convergent are convergent.

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