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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. ... Any series that is not convergent is said to be divergent or to diverge.
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 do not approach zero diverges.
The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.
"Divergence", Encyclopedia of Mathematics, EMS Press, 2001 [1994] The idea of divergence of a vector field; Khan Academy: Divergence video lesson; Sanderson, Grant (June 21, 2018). "Divergence and curl: The language of Maxwell's equations, fluid flow, and more". 3Blue1Brown. Archived from the original on 2021-12-11 – via YouTube
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]
In mathematics, convergence tests are ... is a convergent series, {} is a monotonic sequence ... is a strictly monotone and divergent sequence and the following limit ...
Indeed, the sum of the absolute values of each term is + + + +, or the divergent harmonic series. According to the Riemann series theorem, any conditionally convergent series can be permuted so that its sum is any finite real number or so that it diverges. When an absolutely convergent series is rearranged, its sum is always preserved.
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, ..., which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent.