W4061
... The algebra of sets; ordered sets, the real number system, Euclidean space. Finite, countable, and uncountable sets. Elements of general topology: metric spaces, open and closed sets, completeness and compactness, perfect sets. Sequences and series of real numbers, especially power series; the numbe ...
... The algebra of sets; ordered sets, the real number system, Euclidean space. Finite, countable, and uncountable sets. Elements of general topology: metric spaces, open and closed sets, completeness and compactness, perfect sets. Sequences and series of real numbers, especially power series; the numbe ...
Tiling Proofs of Recent Sum Identities Involving Pell Numbers
... Figure 3: The last fault line occurs at cell k. This fault line occurs at some cell k where 0 ≤ k ≤ 2n − 1. (When k = 0, the only fault line is at “cell 0”, to the left of cell 1.) To the right of this last fault line, A and B consist entirely of dominoes in staggered formation, except for a single ...
... Figure 3: The last fault line occurs at cell k. This fault line occurs at some cell k where 0 ≤ k ≤ 2n − 1. (When k = 0, the only fault line is at “cell 0”, to the left of cell 1.) To the right of this last fault line, A and B consist entirely of dominoes in staggered formation, except for a single ...
4. Techniques of Proof: II
... Two sets S and T are called equinumerous, and we write S ~ T, if there exists a bijective function from S onto T. It is easy to see that “~” is an equivalence relation on any family of sets, and it partitions the family into disjoint equivalence classes. With each equivalence class we associate a ca ...
... Two sets S and T are called equinumerous, and we write S ~ T, if there exists a bijective function from S onto T. It is easy to see that “~” is an equivalence relation on any family of sets, and it partitions the family into disjoint equivalence classes. With each equivalence class we associate a ca ...
Curious and Exotic Identities for Bernoulli Numbers
... i.e., sums related to both the generating functions (A.1) and (A.2). In Sect. A.4 we look at products of Bernoulli numbers and Bernoulli polynomials in more detail. In particular, we prove the result (discovered by Nielsen) that when a product of two Bernoulli polynomials is expressed as a linear co ...
... i.e., sums related to both the generating functions (A.1) and (A.2). In Sect. A.4 we look at products of Bernoulli numbers and Bernoulli polynomials in more detail. In particular, we prove the result (discovered by Nielsen) that when a product of two Bernoulli polynomials is expressed as a linear co ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.