CALC 1501 LECTURE NOTES 4. SEqUEnCEs Definition 4.1. A
... Theorem 4.9 (Monotone Convergence Theorem). Every bounded monotone sequence converges. Proof. Consider the case when {sn } is an increasing sequence bounded above. Since the set S = {sn ; n ∈ N} is bounded, by the Axiom of Completeness, there exists l = sup S. We claim that l is the limit of {sn }. ...
... Theorem 4.9 (Monotone Convergence Theorem). Every bounded monotone sequence converges. Proof. Consider the case when {sn } is an increasing sequence bounded above. Since the set S = {sn ; n ∈ N} is bounded, by the Axiom of Completeness, there exists l = sup S. We claim that l is the limit of {sn }. ...
13.2 Explicit Sequences
... 3. Find the explicit form by plugging in the first term and the common difference 4. Find the function form by simplifying the explicit form 5. To find the specific term replace n with the term you are looking for. ...
... 3. Find the explicit form by plugging in the first term and the common difference 4. Find the function form by simplifying the explicit form 5. To find the specific term replace n with the term you are looking for. ...
real numbers, intervals, and inequalities
... they contain their endpoint, and intervals of the form (a, +⬁) and (−⬁, b) are considered to be open because they do not include their endpoint. The interval (−⬁, +⬁), which is the set of all real numbers, has no endpoints and can be regarded as both open and closed. This set is often denoted by the ...
... they contain their endpoint, and intervals of the form (a, +⬁) and (−⬁, b) are considered to be open because they do not include their endpoint. The interval (−⬁, +⬁), which is the set of all real numbers, has no endpoints and can be regarded as both open and closed. This set is often denoted by the ...
Full text
... n of degree m + l with coefficients involving Bernoulli numbers. See, for example, the papers by Christiano [6] and by de Bruyn and de Villiers [7]. Burrows and Talbot [2] treat this sum as a polynomial in (n + l/2), and Edwards [8] expresses the sums Sm{n) as polynomials in X& and Z £ 2 . Formulas ...
... n of degree m + l with coefficients involving Bernoulli numbers. See, for example, the papers by Christiano [6] and by de Bruyn and de Villiers [7]. Burrows and Talbot [2] treat this sum as a polynomial in (n + l/2), and Edwards [8] expresses the sums Sm{n) as polynomials in X& and Z £ 2 . Formulas ...
Formal Definition of an arithmetic sequence
... Activity IV: [4 minutes] Have the students use the formula to find the nth term of the arithmetic sequences identified in Activity II (pay more attention to exercise # 4) Finding a formula for the sum of the first n terms of any arithmetic sequence through an example (Finite Arithmetic Series): Teac ...
... Activity IV: [4 minutes] Have the students use the formula to find the nth term of the arithmetic sequences identified in Activity II (pay more attention to exercise # 4) Finding a formula for the sum of the first n terms of any arithmetic sequence through an example (Finite Arithmetic Series): Teac ...
Basic Model Theory of Algebraically Closed Fields
... Our starting point will be the following: sometimes two mathematical objects, although nonisomorphic, share exactly the same properties. An example is the famous Lefschetz “transfer principle” which will serve as a guideline until we prove it. Transfer Principle (Lefschetz). Q and C are “the same” w ...
... Our starting point will be the following: sometimes two mathematical objects, although nonisomorphic, share exactly the same properties. An example is the famous Lefschetz “transfer principle” which will serve as a guideline until we prove it. Transfer Principle (Lefschetz). Q and C are “the same” w ...