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The Collaborative International Dictionary
Divergent series

Divergent \Di*ver"gent\, a. [Cf. F. divergent. See Diverge.]

  1. Receding farther and farther from each other, as lines radiating from one point; deviating gradually from a given direction; -- opposed to convergent.

  2. (Optics) Causing divergence of rays; as, a divergent lens.

  3. Fig.: Disagreeing from something given; differing; as, a divergent statement.

    Divergent series. (Math.) See Diverging series, under Diverging.

Wiktionary
divergent series

n. (context mathematics English) An infinite series whose partial sums are divergent

Wikipedia
Divergent series
For an elementary calculus-based introduction, see Divergent series on Wikiversity.

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. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series


$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots =\sum_{n=1}^\infty\frac{1}{n}.$$

The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme.

In specialized mathematical contexts, values can be objectively assigned to certain series whose sequence of partial sums diverges, this is to make meaning of the divergence of the series. A summability method or summation method is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent series


1 − 1 + 1 − 1 + ⋯

the value . Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization.

Usage examples of "divergent series".

The paradoxes of the calculus of small differences, with their tangled substrate of divergent series and asymptotic expansions, were no different to them than were earlier logical worries over differentials, limits, generalized functions, action at a distance, and renormaliza-tion.