Convergence of Series Absolute convergence, Conditional convergence, Examples. Absolute convergence, Conditional convergence, Examples

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  • Convergence of Series Absolute convergence, Conditional convergence, Examples. Absolute convergence, Conditional convergence, Examples.
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  • Absolute convergence In mathematics, a series (or sometimes also an integral) of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite.mathematicsseriesintegralabsolute valuefinite More precisely, a real or complex-valued series is said to converge absolutely if Absolute convergence is vitally important to the study of infinite series because on the one hand, it is strong enough that such series retain certain basic properties of finite sumsthe most important ones being rearrangement of the terms and convergence of products of two infinite seriesthat are unfortunately not possessed by all convergent series. On the other hand absolute convergence is weak enough to occur very often in practice. Indeed, in some (though not all) branches of mathematics in which series are applied, the existence of convergent but not absolutely convergent series is little more than a curiosity. In mathematics, a series (or sometimes also an integral) of numbers is said to converge absolutely if the sum (or integral) of the absolute value of the summand or integrand is finite.mathematicsseriesintegralabsolute valuefinite More precisely, a real or complex-valued series is said to converge absolutely if Absolute convergence is vitally important to the study of infinite series because on the one hand, it is strong enough that such series retain certain basic properties of finite sumsthe most important ones being rearrangement of the terms and convergence of products of two infinite seriesthat are unfortunately not possessed by all convergent series. On the other hand absolute convergence is weak enough to occur very often in practice. Indeed, in some (though not all) branches of mathematics in which series are applied, the existence of convergent but not absolutely convergent series is little more than a curiosity.
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  • More general setting for absolute convergence One may study the convergence of series whose terms a n are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm:abelian topological groupnorm A norm on an abelian group G (written additively, with identity element 0) is a real-valued function on G such that: The norm of the identity element of G is zero: The norm of any nonidentity element is strictly positive: For every x in G, For every x, y in G, Then the function induces on G the structure of a metric space (and in particular, a topology). We can therefore consider G-valued series and define such a series to be absolutely convergent if One may study the convergence of series whose terms a n are elements of an arbitrary abelian topological group. The notion of absolute convergence requires more structure, namely a norm:abelian topological groupnorm A norm on an abelian group G (written additively, with identity element 0) is a real-valued function on G such that: The norm of the identity element of G is zero: The norm of any nonidentity element is strictly positive: For every x in G, For every x, y in G, Then the function induces on G the structure of a metric space (and in particular, a topology). We can therefore consider G-valued series and define such a series to be absolutely convergent if
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  • Relations with convergence If the metric d on G is complete, then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergencea series is convergent if and only if its tails can be made arbitrarily small in normand apply the triangle inequality.complete In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is complete; i.e., a Banach space.Banach space Of course a series may be convergent without being absolutely convergent, the standard example being the alternating harmonic series.alternating harmonic series However, many standard tests which show that a series is convergent in fact show absolute convergence, notably the ratio and root tests. This has the important consequence that a power series is absolutely convergent on the interior of its disk of convergence. It is standard in calculus courses to say that a real series which is convergent but not absolutely convergent is conditionally convergent. However, in the more general context of G-valued series a distinction is made between absolute and unconditional convergence, and the assertion that a real or complex series which is not absolutely convergent is necessarily conditionally convergent (i.e., not unconditionally convergent) is then a theorem, not a definition. This is discussed in more detail below.conditionally convergent If the metric d on G is complete, then every absolutely convergent series is convergent. The proof is the same as for complex-valued series: use the completeness to derive the Cauchy criterion for convergencea series is convergent if and only if its tails can be made arbitrarily small in normand apply the triangle inequality.complete In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is complete; i.e., a Banach space.Banach space Of course a series may be convergent without being absolutely convergent, the standard example being the alternating harmonic series.alternating harmonic series However, many standard tests which show that a series is convergent in fact show absolute convergence, notably the ratio and root tests. This has the important consequence that a power series is absolutely convergent on the interior of its disk of convergence. It is standard in calculus courses to say that a real series which is convergent but not absolutely convergent is conditionally convergent. However, in the more general context of G-valued series a distinction is made between absolute and unconditional convergence, and the assertion that a real or complex series which is not absolutely convergent is necessarily conditionally convergent (i.e., not unconditionally convergent) is then a theorem, not a definition. This is discussed in more detail below.conditionally convergent
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  • Rearrangements and unconditional convergence Given a series with values in a normed abelian group G and a permutation of the natural numbers, one builds a new series, said to be a rearrangement of the original series.permutation A series is said to be unconditionally convergent if all rearrangements of the series are convergent to the same value.unconditionally convergent When G is complete, absolute convergence implies unconditional convergence. Given a series with values in a normed abelian group G and a permutation of the natural numbers, one builds a new series, said to be a rearrangement of the original series.permutation A series is said to be unconditionally convergent if all rearrangements of the series are convergent to the same value.unconditionally convergent When G is complete, absolute convergence implies unconditional convergence.
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  • Theorem Let and let be a permutation of then Proof By the condition of absolute convergence: And now let, and, that is to say the set of all inverse elements of up to N. Substituting into the definition we get: Now we only need a little discussion, why the second definition conforms. Theorem Let and let be a permutation of then Proof By the condition of absolute convergence: And now let, and, that is to say the set of all inverse elements of up to N. Substituting into the definition we get: Now we only need a little discussion, why the second definition conforms.
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  • For this we write qed. The issue of the converse is much more interesting. For real series it follows from the Riemann rearrangement theorem that unconditional convergence implies absolute convergence. Since a series with values in a finite-dimensional normed space is absolutely convergent if each of its one-dimensional projections is absolutely convergent, it follows easily that absolute and unconditional convergence coincide for -valued series.Riemann rearrangement theorem But there is an unconditionally and nonabsolutely convergent series with values in Hilbert space : if is an orthonormal basis, take.Hilbert space Remarkably, a theorem of Dvoretzky-Rogers asserts that every infinite-dimensional Banach space admits an unconditionally but non-absolutely convergent series. For this we write qed. The issue of the converse is much more interesting. For real series it follows from the Riemann rearrangement theorem that unconditional convergence implies absolute convergence. Since a series with values in a finite-dimensional normed space is absolutely convergent if each of its one-dimensional projections is absolutely convergent, it follows easily that absolute and unconditional convergence coincide for -valued series.Riemann rearrangement theorem But there is an unconditionally and nonabsolutely convergent series with values in Hilbert space : if is an orthonormal basis, take.Hilbert space Remarkably, a theorem of Dvoretzky-Rogers asserts that every infinite-dimensional Banach space admits an unconditionally but non-absolutely convergent series.
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  • Products of series The Cauchy product of two series converges to the product of the sums if at least one of the series converges absolut