Convergent series
Conditional and absolute convergence

For any sequence { a 1, a 2, a 3, … } {\displaystyle \left\{a_{1},\ a_{2},\ a_{3},\dots \right\}}, a n ≤ | a n | {\displaystyle a_{n}\leq \left|a_{n}\right|} for all n. Therefore,

This means that if ∑ n = 1 ∞ | a n | {\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|} converges, then ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} also converges (but not vice versa).

If the series ∑ n = 1 ∞ | a n | {\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|} converges, then the series ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} is absolutely convergent. The Maclaurin series of the exponential function is absolutely convergent for every complex value of the variable.

If the series ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} converges but the series ∑ n = 1 ∞ | a n | {\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|} diverges, then the series ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} is conditionally convergent. The Maclaurin series of the logarithm function ln (1 + x) {\displaystyle \ln(1+x)} is conditionally convergent for x = 1.

The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
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