Infinite series

Absolute and conditional convergence

A series ∑∞𝑘=1𝑎𝑘 in a normed vector space 𝑉 converges absolutely iff the series ∑∞𝑘=1‖𝑎𝑘‖ converges. anal

Real or complex

Given any series ∑∞𝑛=1𝑎𝑛, the series is said to be absolutely convergent iff

∞∑𝑛=1|𝑎𝑛|

converges. It can be shown that whenever a series is absolutely convergent the series itself is convergent. However, as can be seen with the Alternating series test, some absolutely divergent series are convergent, and these are said to be conditionally convergent. The Ratio test for absolute convergence is a valuable tool for showing a series is absolutely convergent.

Terminology

• conditionally convergent ⟹ convergent, but absolutely divergent
• absolutely convergent ⟹ convergent
• divergent ⟹ absolutely and conditionally divergent

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