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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 .
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 the limit superior is allowed to take the value .
Ratio test: Let be a series with for all sufficiently large , and define
In particular, if the ratio has a limit , the series converges absolutely for , diverges for , and the test is inconclusive for .