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), …

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