In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted
The nth partial sum Sn is the sum of the first n terms of the sequence; that is,
A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,
If the series is convergent, the (necessarily unique) number is called the sum of the series.
The same notation
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b.
Any series that is not convergent is said to be divergent or to diverge.
The reciprocals of the positive integers produce a divergent series (harmonic series):
Alternating the signs of the reciprocals of positive integers produces a convergent series (alternating harmonic series):
The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes):
The reciprocals of triangular numbers produce a convergent series:
The reciprocals of factorials produce a convergent series (see e):
The reciprocals of square numbers produce a convergent series (the Basel problem):
The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
The reciprocals of powers of any n>1 produce a convergent series:
Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
Convergence tests
There are a number of methods of determining whether a series converges or diverges.
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In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. A series of the form where either all an are positive or all an are negative, is called an alternating series.
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing the convergence or divergence of an infinite series or an improper integral. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known.
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. Suppose and are two sequences. Then, Using the forward difference operator , it can be stated more succinctly as Summation by parts is an analogue to integration by parts: or to Abel's summation formula: An alternative statement is which is analogous to the integration by parts formula for semimartingales.
Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond
Concepts de base de l'analyse réelle et introduction aux nombres réels.
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