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Concept
Convergent series
Summary
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.
Official source
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