: Mathématiques: divergent series

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divergent series

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

${\\displaystyle 1+{\ rac {1}{2}}+{\ rac {1}{3}}+{\ rac {1}{4}}+{\ rac {1}{5}}+\\cdots =\\sum _{n=1}^{\\infty }{\ rac {1}{n}}.}$

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