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Concept
Convergence tests
Summary
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .
If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.
This is also known as d'Alembert's criterion.
Suppose that there exists such that
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
This is also known as the nth root test or Cauchy's criterion.
Let
where denotes the limit superior (possibly ; if the limit exists it is the same value).
If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.
The series can be compared to an integral to establish convergence or divergence. Let be a non-negative and monotonically decreasing function such that . If
then the series converges. But if the integral diverges, then the series does so as well.
In other words, the series converges if and only if the integral converges.
A commonly-used corollary of the integral test is the p-series test. Let . Then converges if .
The case of yields the harmonic series, which diverges. The case of is the Basel problem and the series converges to . In general, for , the series is equal to the Riemann zeta function applied to , that is .
If the series is an absolutely convergent series and for sufficiently large n , then the series converges absolutely.
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