Theorem: Binomial Theorem edit contribute as guest [id:1397] For all natural numbers \(n\in\mathbb N\) and any two elements \(x,y\in R\) of a ring \((R,+,\cdot)\), there is a closed formula for the sum
Equivalently, \((x+y)^n\) can be expanded to the sum \(\sum_{k=0}^n{n\choose k}x^{n-k}y^k\). The symbol \({n \choose k}\) denotes the binomial coefficients .
Further Reading
[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983
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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 841: algebraic structure of natural numbers together with addition 708: alphabet, letter, concatenation, string, empty string, formal language 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 993: binomial coefficients 837: cancellation property 980: cardinal number 748: cartesian product 553: commutative (abelian) group 706: commutative monoid 984: comparing cardinal numbers 1547: comparing natural numbers using the concept of addition 982: counting the sets elements using its partition 1539: equality, inequality 978: equipotent sets 574: equivalence relation, equivalent classes, partitions, representative elements, quotient sets 1455: existence of natural zero (neutral element of addition of natural numbers) 988: finite cardinal numbers and set operations 985: finite set, infinite set 671: group 836: groupoid (magma) 661: identity, neutral element, left identity, right identity 769: injective function 705: law of excluded middle 659: monoid 714: negation 697: order relation for natural numbers 747: ordered pair, n-tuple 723: ordinal number 504: peano axioms 1308: properties of binary relations between two sets 721: properties of transitive sets 572: reflexive, symmetric and transitive relation 710: sematics, proposition 660: semigroup 707: set of binary logical values (true and false) 664: set of natural numbers (peano) 550: set, set element, empty set 719: set-theoretic definition of order relation for natural numbers 552: subset, superset, union, intersection, set difference, set complement, power set 986: subsets of finite sets 774: successor of oridinal 770: surjective function 744: the proving principle by contradiction 592: total maps (functions) 720: transitive set 698: well-ordering principle 1427: zermelo-fraenkel axioms
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