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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. If 1 < p, then the nth-term test is inconclusive, but the series is convergent by the integral test for convergence.
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.
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 ...
where is the number of terms in the progression and is the common difference between terms. The formula is essentially the same as the formula for the standard deviation of a discrete uniform distribution , interpreting the arithmetic progression as a set of equally probable outcomes.
First, the magnitude of all the nth roots is the nth root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the n th roots is θ / n {\displaystyle \theta /n} , where θ {\displaystyle \theta } is the angle defined in the same way for the number whose root is ...
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
The Cauchy formula is generalized to non-integer parameters by the Riemann–Liouville integral, where is replaced by , >, and the factorial is replaced by the gamma function.
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series =, where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.