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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 denotes the absolute value of the term.
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 .
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.
An absolute value is the function defined by
An absolutely convergent series is a series such that the series of absolute values is a convergent series.
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.
A power series is a series of the form with real coefficients and a real center .
There exists a number , called the radius of convergence, such that the series converges for and diverges for (the endpoint behavior at is decided separately). The radius can often be computed using the Cauchy–Hadamard theorem.
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