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The more general class of p-series, =, exemplifies the possible results of the test: If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.
2 Examples. 3 Convergence of products. ... This is also known as the nth-term test, ... be an infinite series with real terms and let : ...
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. Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
The n th term describes the length of the n th run ... At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence.
A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients , because the coefficients of the linear function (1 and 1) are constants that do not depend on n . {\displaystyle n.}
The harmonic series is the infinite series = = + + + + + in which the terms are all of the positive unit fractions. It is a divergent series : as more terms of the series are included in partial sums of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit.
The infinite series whose terms are the natural numbers 1 ... The nth partial sum of the series is the ... The latter series is an example of a Dirichlet ...
The sequence 0, 3, 8, 15, ... is formed according to the formula n 2 − 1 for the nth term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, (sequence A000396 ...