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A necessary condition for a series to converge is that its terms tend to zero.
Terms go to zero: If the series converges, then
Equivalently (the divergence test), if does not tend to , then the series is divergent. This is a basic prerequisite for many convergence tests and is often used to quickly rule out convergence before applying finer criteria like the root test or ratio test.
A series is a formal expression together with the associated partial sums .
Whether a series has a value is determined by the behavior of its partial sums: it either converges or diverges. Many standard tools for deciding this are collected as convergence tests such as the ratio test and the comparison test.
A convergent series is a series whose partial sums converge to a finite limit , in which case one writes .
A divergent series is a series whose partial sums do not converge to a finite limit, i.e. it is not a convergent series.
Divergence may happen because partial sums grow without bound or because they oscillate. A basic necessary condition for convergence is terms go to zero, whose contrapositive often certifies divergence quickly.
Root test: For a series , define
where denotes the absolute value of the term.
Ratio test: Let be a series with for all sufficiently large , and define