RI Chapter 8 Sequences

RAFFLES INSTITUTION H3 Mathematics 9820 _____________________________________________________________________________________________ _________________________ Chapter 8: Sequences and Series Page 1 of 24 Chapter 8 Sequences and Series This chapter is concerned with the study of sequences and series. The material has far -reaching applications in various areas of science and engineering and is the cornerstone for many branches of mathematics. We proceed via sequences , before looking at series as a sequence of partial sums. SYLLABUS INCLUDES  Sequences and series, general terms, sums, limiting behaviours and bounds CONTENT 1 Sequences 1.1 Limit of a sequence 1.2 Monotone Sequences 1.3 Recurrence Relations 2 Series 2.1 Computing Partial Sums 2.2 Tests of Convergence

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 8: Sequences and Series Page 2 of 24 1 Sequences In everyday usage, the term “sequence” suggests a succession of objects or events in a specified order. Informally, a “sequence” in mathematic s refers to an unending succes sion of numbers. Some examples are 11 23 1, 2,3,... 2, 4,6,... 1, , ,... 1, 1,1, 1,... In each case, ellipses (the three dots) are used to suggest that the sequence continu es indefinitely, following the (hopefully) obvious pattern. The numbers in the sequence are called the terms of the sequence, and are described according to the positions they occupy. Thus we speak of the first term, second term and so forth of a sequence. The most common way to specify a sequence is to give a formula for the terms, telling us how to compute the nth term in terms of n. For the second example above, we see that the nth is given by the formula 2n. This can be expressed using the bracket notation by writing   12 nn   . When we want to talk about a sequence without actually specifying the numerical values of the terms, we use a letter with subscripts to denote the terms:    12 1, ,..., ,... or or .n n n na a a a a   Let us now consider the idea of a sequence more precisely. Notice that in bracket notation, such as {2n}, we are actually specifying a rule that tells us how to associate a numerical value, 2 n in this case, with each positive integer. We have a name for such an as sociation – a function! Thus we have the following formal definition. Definition 1. A sequence is a function whose domain is the set of positive integers. The codomain is intentionally left unspecified above to allow for the possibility of sequences of other entities (like complex numbers), but c an be taken to be the real num bers for this chapter. Thus a sequence is a function f:   . The notation we have been using is then simply a listing of the functional values f(1), f(2), …, f(n), …

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 8: Sequences and Series Page 3 of 24 Thinking of a sequence as a function also leads us to inquire about the graph of a sequence. Since the domain is  , the graph of a sequence consists of a succession of isolated points. For example, the graphs of  1 n and  ( 1) n are as follows: One of the questions we can ask about a sequence is its behaviour for large values of n. This is what we consider in the next section and the graph of a sequence can be a useful vi sual aid in this pursuit. 1.1 Limit of a Sequence The three sequences {n},  1 n and  ( 1) n behave very differently as n gets larger and n larger. In the sequence { n} the terms increase without bound; in the sequence  1 n the terms decrease and approaches a “limiting” value of 0; and in the sequence  ( 1) n the terms oscillate between 1 a nd −1. It is the behaviour of  1 n that we are most interested in, but before making the concept precise, let us consider some more examples. Exercise 1 Describe the behaviour of the following sequences, with the aid of your GC if necessary (i) 3 231 n n   ; (ii) 1 n n   ; (iii) sin n n  ; (iv)  sin n .

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 8: Sequences and Series Page 4 of 24 Let us try to make the term “limit” pre cise. To say that the sequence  na approaches a limit A as n gets large is meant to convey the idea that, eventually, the terms in the sequence become arbitrarily close to the number A. This means that, whatever our criterion for closeness is (such as being within a distance of ε of A), all the terms in the sequence from some point onwards will be within that distance of A. Geometrically, if we sketch the lines y = A + ε and y = A − ε on the graph of the sequence, all the terms in the sequence after some point, say N, will be trapped within the band between these lines. This is illustrated in the diagram below: If a sequence  na has a limit A, we say that the sequence converges to A and write lim nn aA   A sequence that does not have a finite limit is said to diverge. Example 1 Let us look at some common sequences and see if they converge to a finite limit. The aim is to develop an intuitive feel for the idea without getting into the details of the proofs. (i) A constant sequence, that is, a sequence in which each t erm na is a fixed number, say a, converges and has the limit a. (ii) The sequence  1 n converges to the limit 0. (iii) The sequences {n} and {2n} increases without bound and diverges. The sequence   2n also diverges, but does so by decreasing without bound.

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 8: Sequences and Series Page 5 of 24 (iv) The sequences 1 n n   and    1 21 n  both converge to 1. The terms of 1 n n   are always less than 1 and approache s 1 from below, while those of    1 21 n  are alternately below or above 1, but nevertheless still approach 1. (v) The terms of the sequence  ( 1) n oscillates alternately between −1 and 1 and does not converge. The sequence  sin n also does not converge, but its terms vary erratically between −1 and 1. (vi) Finally, the sequence sin n n  converges to 0 even though its terms flips be tween being positive or negative erratically. We will not attempt to formally define what it means for a sequence to converge or prove the following properties; this you will do in a course on analysis. Instead we will just list here some results that will allows us to determine if a sequence converges and find its limit when it does. Theorem 1 If a sequence is convergent, the limit is unique. Also, every subsequence is convergent and has the same limit. A subsequence of  na refers to a sequence obtained by picking some of the terms of  na . For example, two subsequences of 1, 2, 3, 4, 5, ..., n, ... are 1, 2, 4, 6, ..., 2n, ... and 1, 3, 9, 27, 81, ..., 3n, .... Similar to our discussion in inequalities, we will also introduce the notion of bounds for sequences. A sequence  na is said to be bounded above if there is a real number M such that all its terms are less than M. In other words, naM for all n. Similarly,  na is said to be bounded below if there is a real number L such that naL for all n. Finally,  na is said to be bounded if it is both bounded above and bounded below. Since the terms of a convergent sequence must eventually lie close to its limit, it is not hard to see that the