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A convergence test using the limsup of the nth roots of the term magnitudes.
Root test. For a series , define
where the limit superior is allowed to take the value .
The root test is especially natural for power series and is closely related to the Cauchy–Hadamard theorem; compare also the ratio test.
A series is a formal expression together with the associated partial sums .
A limit superior of a real sequence is the extended real number
provided the right-hand side is interpreted in .
An absolutely convergent series is a series such that the series of absolute values is a convergent series.
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.
Cauchy–Hadamard theorem: For a power series (real or complex), define
Set the radius of convergence
with the conventions and . Then:
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 .