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(1) Use the monotonic convergence theorem to prove the sequences converge. (a) an = 4n?3. 2n. Solution: First, we need to determine if this is an increasing or decreasing se- quence. We look to an+1 = 4n+1. 2n+1 and an = 4n?3. 2n. = 2(4n?3). 2n+1 . For n large, an > an+1. Therefore, it is a decreasing sequence.
21 Jun 2011 Monotone Sequences. Maybe you remember hearing about how the real numbers can be broken up into two types of numbers. There are algebraic numbers such as. 1,3,. v. 2,1/. v. 2,. v. 2 +. v. 3 i.e., numbers from some sort of algebraic manipulation and then there are transcendental numbers. I really
Mathematical analysis depends on the properties of the set R of real numbers, so we should begin by saying something about it. There are two familiar ways to represent real numbers. Geometrically, they may be pictured as the points on a line, once the two reference points correspond- ing to 0 and 1 have been picked.
This will be the usual situation when we get to infinite series. The definition of limit implies that we must know the limit L before we prove that L is the limit, but if we look at certain types of sequences, we can show that a limit exists without knowing what it is.
Calculus. Monotonic sequences www.maths.nuigalway.ie/MA1802. Niall Madden. (Niall.Madden@NUIGalway.ie). Lecture 10: Tuesday, 5th February 2013. 1 Recall: Bounded Sequences. 2 Increasing and decreasing sequences. 3 The Monotone Convergence Theorem. 4 An Example. 5 2 3: Introduction to Infinite Series.
Module 1 : Real Numbers, Functions and Sequences. Lecture 3 : Monotone Sequence and Limit theorem. [ Section 3.1 : Monotone Sequences ]. Objectives. In this section you will learn the following. The concept of a sequence to be monotonically increasing/ decreasing. Convergence of monotone sequences.
If a sequence is monotone nondecreasing it satisfies one of the convergence properties below. 1. The monotone convergence theorem tells us the sequence converges to a limit 1 because the sequence is . Be sure to see the handout on factorials and bounds. voyager.dvc.edu/~LMonth/Calc2/HdBounds.pdf.
7.4 Bounded Sequences converge. Theorem. Let (an) be a bounded above monotone non-decreasing sequence. Then (an) is convergent. Proof. Let l = sup{an : n ? N}; exists by C. Let ? > 0. Then l??<l, so is not sup. Hence for some N ? N, l ? ? l.
1. Lecture 2 : Convergence of a Sequence, Monotone sequences. In less formal terms, a sequence is a set with an order in the sense that there is a first element, second element and so on. In other words for each positive integer 1,2,3, , we associate an element in this set. In the sequel, we will consider only sequences of
Sequences: Convergence and Divergence. In Section 2.1, we consider (infinite) sequences, limits of sequences, and bounded and monotonic sequences of real numbers. In addition to certain basic properties of convergent sequences, we also study divergent sequences and in particular, sequences that tend to positive or
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