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If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute
A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒ n (x) = x n /n on the open interval (−1,0), then the series Σf n (x) converges uniformly by comparison of the partial sums to those of Σ(−1) n /n, and the series Σ|f n (x)| converges absolutely at each point by the geometric series test, but Σ|f n (x)| does ...
In a normed vector space, one can define absolute convergence as convergence of the series (| |). Absolute convergence implies Cauchy convergence of the sequence of partial sums (by the triangle inequality), which in turn implies absolute convergence of some grouping (not reordering). The sequence of partial sums obtained by grouping is a ...
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
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. Agnew's theorem characterizes rearrangements that preserve convergence for all series.
For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number (see the article on the Riemann rearrangement theorem for further discussion). An example of a convergent series is a geometric series which forms the basis of one of Zeno's famous paradoxes:
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