Sequences and Summations: Lecture Notes for Math 6A, Spring Quarter 2003-04, Study notes of Linear Algebra

Download Sequences and Summations: Lecture Notes for Math 6A, Spring Quarter 2003-04 and more Study notes Linear Algebra in PDF only on Docsity! Jim Lambers Math 6A Spring Quarter 2003-04 Lecture 20 Notes These notes correspond to Section 3.2 in the text. Sequences and Summations Sequences When working with a set, it is often helpful to describe the elements of a set by numbering them. For instance, a set with four elements can be described using the notation {a1, a2, a3, a4}. Such a numbering can be used to provide a concise description of all of the elements of the set, which facilitates use of the set in the context of solving problems. As numbering elements effectively imposes an order on the elements, we can view the numbered elements as forming a sequence. We now define this concept precisely. Definition (Sequence) A sequence is a function whose domain is a subset of Z, the set of integers. Each element of the range of the sequence is called a term of the sequence. For each integer n in the domain of the sequence, the image of n, is denoted by an. Remark Although a sequence is defined as a function, it is usually identified with its range, which is the set of all of its terms. That is, a sequence with elements {ak, ak+1, ak+2, . . . , am} can be described using the notation {an}mn=k. The notation {an} is also used when the domain of the sequence is understood from the contest. We illustrate this identification of sequences in the following examples. 2 The following two types of sequences are of particular interest. Definition (Geometric Progression) A geometric progression is a sequence whose terms have the form a, ar, ar2, . . ., arn, where a is a real number called the initial term and r is real number known as the common ratio. Definition (Arithmetic Progression) An arithmetic progression is a sequence whose terms have the form a, a + d, a + 2d, . . ., a + nd, where a is a real number called the initial term and d is real number known as the common difference. Example Let a = 1 and r = −1/2. Then the geometric progression with initial term a and common ratio r contains the elements {1,−1/2, 1/4,−1/8, 1/16, . . .}. This sequence can be described more concisely using the notation {(−1/2)n}∞n=0, or simply {(−1/2)n} when the length of the sequence is understood from the context. 2 1 Example Suppose that a sequence beginning with the number 4 has elements that are spaced 5 apart; that is, the sequence has the elements {4, 9, 14, 19, 24, 29, . . .}. We can then recognize that this sequence is an arithmetic progression with initial term 4 and common difference 5. This sequence can be represented more concisely using the notation {4 + 5n}. 2 Summations Often it is necessary to determine the sum of the elements of a sequence {an} of numbers. This may seem like a trivial task involving nothing more than the arithmetic of adding the numbers, but this approach is not feasible if the sequence has infinitely many elements. Even if the sequence is finite, it may be possible to obtain a formula for the sum that is much easier to compute. We have previously used the notation n∑ i=m an to represent the sum am + am+1 + · · ·+ an. This notation is called summation notation, or sigma notation, due to the use of the Greek letter sigma. The variable i is called the index of summation. The number m is called the lower limit and the number n is called the upper limit. The index of summation assumes the values of each integer between the lower limit and the upper limit, and the sum includes all terms in the sequence {an} that are images of integers in this range. Sums of elements of sequences are known as series, which we now define precisely for both finite and infinite sequences. Definition (Series) Let {an} be a sequence. The sum n∑ i=m ai, for any integers m and n, is called a series. If the domain of {an} is the set of all integers not less than m, then the sum ∞∑ i=m ai represents the sum of all terms of the sequence. This sum is called an infinite series. Each term ai of the sequence {an} is called a term of the series. Example Let {an}∞n=0 be a sequence in which each term an has the form xn/n!, where x is a real number. Then the series ∞∑ n=0 xn n! is an infinite series. The sum of this series is exp(x), or ex, which is the value of the natural exponential function at x. 2 2