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A series is convergent (or converges) if and only if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number.
N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers. In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms ...
A famous example of an application of this test is the alternating harmonic series = + = + +, which is convergent per the alternating series test (and its sum is equal to ), though the series formed by taking the absolute value of each term is the ordinary harmonic series, which is divergent.
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
2 Examples. 3 Convergence of products. 4 ... is a convergent series, {} is a monotonic sequence ... is a strictly monotone and divergent sequence and the following ...
The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space. 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.
We say that "the limit of the sequence equals ." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). [1] If such a limit exists and is finite, the sequence is called convergent. [2] A sequence that does not converge is said to be divergent. [3]
[4] [10] This is because if Σa n is convergent, a second convergent series Σb n can be found which converges more slowly: i.e., it has the property that lim n->∞ (b n /a n) = ∞. Furthermore, if Σa n is divergent, a second divergent series Σb n can be found which diverges more slowly: i.e., it has the property that lim n->∞ (b n /a n) = 0.