Proposition: Direct Comparison Test For Absolutely Convergent Series (Majorant Criterion) edit contribute as guest [id:1270] Given an infinite series \(\sum_{k=0}^\infty x_k\), we want to test, if it is an absolutely convergent series . A sufficient condition for this is to find a series \(\sum_{k=0}^\infty y_k\) with the following properties:
\(|x_k|\le y_k\) for all \(k\), and
\(\sum_{k=0}^\infty y_k\) is a convergent series .
A series \(\sum_{k=0}^\infty y_k\) with the above properties is called the majorant of the series \(\sum_{k=0}^\infty x_k\).
References
[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983
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1081: absolute value of rational numbers 583: absolute value of real numbers 890: addition of integers 842: addition of natural numbers 1428: addition of natural numbers is associative 1432: addition of natural numbers is cancellative 1430: addition of natural numbers is commutative 892: algebraic structure of integers together with addition and multiplication 841: algebraic structure of natural numbers together with addition 877: algebraic structure of natural numbers together with multiplication 1108: all cauchy sequences converge in the set of real numbers (completeness principle) 31: associative law of addition 668: axiom of associativity 672: axiom of commutativity 682: axiom of distributivity 669: axiom of existence of an identity 670: axiom of existence of inverse elements 771: bijective function 571: binary relations 837: cancellation property 748: cartesian product 1072: cauchy sequence 553: commutative (abelian) group 33: commutative law 38: commutative law of multiplication 706: commutative monoid 880: commutative ring 1103: commutative semigroup 888: construction of fields from integral domains 839: construction of groups from commutative and cancellative semigroups 141: convergent sequences 148: convergent sequences and limits 827: cosets 844: definition of integers 1033: definition of rational numbers 1109: definition of real infinite series, partial sums 1105: definition of real numbers 574: equivalence relation, equivalent classes, partitions, representative elements, quotient sets 35: existence of negative numbers (inverse elements of addition) 40: existence of one (neutral element of multiplication) 34: existence of zero (neutral element of addition) 191: factor groups 274: factor rings 557: field 671: group 679: group homomorphism 836: groupoid (magma) 401: homomorphism 1062: ideal 661: identity, neutral element, left identity, right identity 769: injective function 575: irreflexive, asymmetric and antisymmetric relation 838: isomorphic semigroups 614: metric (distance) 617: metric space 659: monoid 891: multiplication of integers 876: multiplication of natural numbers 1434: multiplication of natural numbers is associative 1440: multiplication of natural numbers is cancellative 1435: multiplication of natural numbers is commutative 273: normal subgroups 849: open ball, neighborhood 1075: order relation for integers 697: order relation for natural numbers 1076: order relation for rational numbers 1107: order relation for real numbers 747: ordered pair, n-tuple 584: ordering axioms 723: ordinal number 504: peano axioms 585: positive and negative numbers 576: preorder (quasiorder), partial and total order, poset and chain 1308: properties of binary relations between two sets 829: properties of cosets 619: properties of the absolute value 721: properties of transitive sets 875: real sequence 572: reflexive, symmetric and transitive relation 1436: right-distributivity law for natural numbers 683: ring 885: ring homomorphism 594: rules of calculation with inequalities 660: semigroup 874: sequence 664: set of natural numbers (peano) 550: set, set element, empty set 718: set-theoretic definitions of natural numbers 887: subfield 554: subgroup 552: subset, superset, union, intersection, set difference, set complement, power set 774: successor of oridinal 770: surjective function 1090: the absolute value makes the set of rational numbers a metric space. 618: the distance of real numbers makes real numbers a metric space. 1030: the distributivity law for natural numbers 657: the proving principle of complete induction (variant 1) 1101: the unit ring of all rational cauchy sequences 592: total maps (functions) 720: transitive set 43: uniqueness of 0 48: uniqueness of 1 50: uniqueness of negative numbers 821: unit and unit group, zero divisor and integral domain 1427: zermelo-fraenkel axioms 879: zero ring
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