MATH 142 - Infinite Series Joe Foster Definitions: Given a sequence of numbers {a n} ∞ n=1, an expression of the form X∞ n=1 a n = a1 +a2 +a3 +··· +a n +··· is an infinite series. The number a n is the nth term of the series. The sequence {Sn} ∞ n=1 defined by S n:= Xn n=1 a n = a1 +a2 +a3 +··· +a n is called the sequence of partial sums of the series, the number S n being ...
INFINITE SERIES 2.1 Sequences: A sequence of real numbers is defined as a function ,where is a set of natural numbers and R is a set of real numbers. A sequence can be expressed as or is a sequence. For example Convergent sequence: A sequence converges to a number l, if
1/8, etc. Since the sequence is infinite, the distance cannot be traveled. Remark. The steps are terms in the sequence. ˆ 1 2, 1 4, 1 8, ˙ Sequences of values of this type is the topic of this first section. Remark. The sum of the steps forms an infinite series, the topic of Section 10.2 and the rest of Chapter 10. 1 2 + 1 4 + 1 8 ...
12 INFINITE SEQUENCES AND SERIES 12.1 SEQUENCES SUGGESTED TIME AND EMPHASIS 1 class Essential material POINTS TO STRESS 1. The basic definition of a sequence; the difference between the sequences {an} and the functional value f (n). 2. The meanings of the terms “convergence” and “the limit of a sequence”. 3.
INFINITE SEQUENCES AND SERIES MIGUEL A. LERMA 1. Sequences 1.1. Sequences. An in nite sequence of real numbers is an ordered unending list of real numbers. E.g.: 1;2;3;4;::: We represent a generic sequence as a1;a2;a3;:::,anditsn-th as a n. In order to de ne a sequence we must give enough information to nd its n-th term. Two ways of doing this ...
nite series and sequences in engineering and the physical sciences. The material has been carefully selected to help the reader develop the techniques needed to con dently utilize in nite series. The book begins with in nite series and sequences before moving onto power series, complex in nite series, and nally Fourier, Legendre, and Fourier ...
Theorem (Monotonic Sequence Theorem). Everybounded, monotonic sequence is convergent. The geometric series X∞ n=1 arn−1 = a+ar +ar2 +··· is convergent if |r| < 1, and its sum is a 1−r. If |r| ≥ 1, the geometric series is divergent. The Test for Divergence. Iflim n→∞ a n doesnotexistoriflim n→∞ a n 6= 0, then the series P ∞ ...
review of sequences and infinite series 467 On the other hand, the sequence defined by an = 1 2n approaches a limit as n gets large. This is depicted in FigureA.2. Another re-lated series, an = ( 1)n 2n, is shown in FigureA.3. This sequence is the alternating sequence f 1 2, 1 4, 1 8,. . .g. Definition A.3. The sequence an converges to the ...
symbols, we say that a sequence is a function f:N→ R. Sequences are written in a few different ways, all equivalent; these all mean the same thing: a1,a2,a3,... {a n} ∞ n=1 {f(n)}∞ n=1 As with functions on the real numbers, we will most often encounter sequences that can be expressed by a formula. We have already seen the sequence a
sequence. The function is defined only on integer inputs, and the graph consists of some points in the xy -plane, located at (2, a 2),….. , (n, an ), ….. DEFINITION Infinite Sequence. An infinite sequence of numbers is a function whose domain is the set of positive integers.
4 1 Infinite Sequences and Series With the inclusion of the negative integers, the equation p+x = q (1.6) has a unique integer solution x(≡ q−p) for every pair of integers p,q.Properties (1.2)–(1.5) characterize the integers as a group Z under addition, with 0 as an identity element. The fact
An important example of an infinite series is the geometric series a + ar + ar2 + ar3 + . . . + arn–1 +. . . = a 0 Each term is obtained from the preceding one by multiplying it by the common ratio r. If r = 1, then s n = a + a +. . . + a = na → Since lim n → s n doesn’t exist, the geometric series diverges in this case.
- The sequence is bounded below if there is a number m such that a n ≥M for all n ≥1 (If a sequence is bounded above and bounded below, we say that the sequence is bounded. If a sequence is not bounded, we say that it is unbounded.) Ex: a) For positive integer n, leta 4 3n 2 n = + − . Show that the sequence {a n}is monotone increasing and ...
Infinite sequences (Sect. 10.1) I Overview: Sequences, series, and calculus. I Definition and geometrical representations. I The limit of a sequence, convergence, divergence. I Properties of sequence limits. I The Sandwich Theorem for sequences. Definition and geometrical representations Definition An infinite sequence of numbers is an ordered set of real numbers.
In particular this is useful for using L’Hôpital’s rule in computing limits of sequences. E.g: − ex ex en = lim = (L’Hôpital’s rule) = lim = ∞. x→∞ x x→∞ 1 n→∞ n lim NOTES ON INFINITE SEQUENCES AND SERIES 3 1.6. Bounded Monotonic Sequences. A monotonic sequence is a sequence that always increases or always decreases.
Chapter 11: Infinite Sequences and Series Section 11.1: Sequences Objective: In this lesson, you learn how to define sequences and determine their convergence or divergence using the Limit Laws, the Squeeze Theorem, boundedness, or monotonicity. I. Infinite Sequences Definition: A sequence A sequence is a list of nnumbers written in a definite ...
NOTES ON INFINITE SEQUENCES AND SERIES OF FUNCTIONS 1. Sequences and Series of Functions 1.1. Pointwise and Uniform convergence of functions. Let ff ngbe a sequence of functions de ned on a subset EˆC. We say that the sequence converges pointwise to a function gif for every z2E, lim n!1f n(z) = g(z). Precisely, this means:
Another very important series is logarithmic series which is also in the form of infinite series. We state the following result without proof and illustrate its application with an example. Theorem If | x | < 1, then ( ) 2 3 log 1 ... e 2 3 x x + = − + −x x The series on the right hand side of the above is called the logarithmic series.
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