Infinite sequences

(bq) part 2 book "calculus early transcendentals" has contents: parametric equations and polar coordinates, infinite sequences and series, vectors and the geometry of space, vector functions, partial derivatives, vector calculus, second order differential equations.
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(bq) part 2 book "3,000 solved problems in calculus" has contents: exponential growth and decay, inverse trigonometric functions, integration by parts, trigonometric integrands and substitutions, improper integrals, planar vectors, polar coordinates, infinite sequences, partial derivatives,...and other contents.
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(bq) part 1 book "advanced calculus with applications in statistics" has contents: an introduction to set theory, basic concepts in linear algebra, limits and continuity of functions, differentiation, infinite sequences and series, integration, multidimensional calculus.
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5.3. The Integral and Comparison Test
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5.1.1 Limits of Sequences  A sequence is an infinite ordered list of numbers where each term is obtained according to a fixed rule.  Symbolically the terms of a sequence are represented with indexed letters a1, a2, a3, …, an , … a1 is the first term, a2 is the second term,… an is the nth term…
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An alternating series is a series whose terms are alternately positive and negative. By the Monotonic Sequence Theorem, the increasing bounded above sequence s2 s3 s5 … converges to a limit s‘ s.
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We can add the terms of a sequence {an } and get an expression of the form: a1+ a2+ a3+ …+ an + … which is called a series and denoted by However what does it mean by the sum of infinitely many terms? Example. We can try to add the terms of the series 1+2+3+…+n+… and get the cumulative sums 1, 3, 6, 10, …, The nth sum n(n+1)/2 becomes very large as n increases
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In this paper, we study the convergence for martingale sequences of random bounded linear operators. The condition for the existence of such a infinite product of random bounded linear operators is established.
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In this article, we investigate a risk model with a quota(α, β) reinsurance contract. The premium process and claim process are assumed to be independent sequences of indentically distributed random variables. Using inductive method, we obtain upper bound of infinitetime ruin probability of an insurance company.
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The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition, as in the first five editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement.
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Chapter 1 PROBABILITY DISTRIBUTIONS ON THE REAL LINE INFINITELY DIVISIBLE LAWS This chapter is of an introductory nature, its purpose being to indicate some concepts and results from the theory of probability which are used in later chapters .
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Chapter 15 APPROXIMATION OF DISTRIBUTIONS OF SUMS OF INDEPENDENT COMPONENTS BY INFINITELY DIVISIBLE DISTRIBUTIONS 1 . Statement of the problem We here consider the general problem of the limiting behaviour of the distribution function F§ (x) of the sum (15.1 .1) of independent random variables with the same distribution F
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We prove that if A is a subset of at least cn1/2 elements of {1, . . . , n}, where c is a suﬃciently large constant, then the collection of subset sums of A contains an arithmetic progression of length n. As an application, we conﬁrm a long standing conjecture of Erd˝s and Folkman on complete sequences. o
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(bq) part 2 book "a transition to advanced mathematics" has contents: infinite sets, countable sets, the ordering of cardinal numbers, comparability of cardinal numbers and the axiom of choice, algebraic structures, operation preserving maps, completeness of the real numbers, the bounded monotone sequence theorem,...and other contents.
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The resulting graph is a nucleus for constructing models of possible worlds in which the sentence is true. • Laws of the world behave like demons or triggers thai monitor the models and block illegal extensions. • If a surface model could be extended infinitely deep, the result would be a complete standard model. This approach leads to an infinite sequence of algorithms ranging from plausible inference to exact deduction; they are analogous to the varying levels of search in game playing programs. ...
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A problem being presented by us in [4] was "Is the Theorem 1.2 true for infinite topological spaces?". In this paper, we will solve the above problem by proving that Rational Squence Topological space is a T1space, that has πgpregularity but also non πgpnormality.
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RealTime Digital Signal Processing  Chapter 1: Introduction to RealTime Digital Signal Processing
Signals can be divided into three categories ± continuoustime (analog) signals, discretetime signals, and digital signals. The signals that we encounter daily are mostly analog signals. These signals are defined continuously in time, have an infinite range of amplitude values, and can be processed using electrical devices containing both active and passive circuit elements. Discretetime signals are defined only at a particular set of time instances. Therefore they can be represented as a sequence of numbers that have a continuous range of values. ...
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Introduction to RealTime Digital Signal Processing Signals can be divided into three categories ± continuoustime (analog) signals, discretetime signals, and digital signals. The signals that we encounter daily are mostly analog signals. These signals are defined continuously in time, have an infinite range of amplitude values, and can be processed using electrical devices containing both active and passive circuit elements. Discretetime signals are defined only at a particular set of time instances.
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