Math 2414 Section 8 2 Sequences Sequences In mathematics the word sequence is employed to convey the idea of a set of things arranged in order That is a sequence is a list of numbers Each number in the sequence is called a term If for every positive integer n in other words the natural numbers there is associated a real or a complex number n then the ordered set a1 a2 a3 K an K is said to define an infinite sequence The important thing here is that each member of the set a a has been labeled with an integer so that we may speak of the first term 1 the second term 2 a a a and in general the nth term n Each term n has a successor n 1 and hence there is no last term The most common examples of sequences can be constructed if we give some rule or formula for a 1 n defines a sequence whose first describing the nth term Thus for example the formula n 1 1 1 1 1 five terms are 2 3 4 5 Sometimes two or more formulas may be employed as for example a2 n 1 1 a2 n 2n 2 The first six terms in this case are Another common way to define a sequence is by a set of instructions which explains how to carry on after a given start Thus we may have a1 a2 1 an 1 an an 1 for n 2 The first six terms of this famous sequence whose terms are called Fibonacci numbers are In any sequence the essential thing is that there be some function f defined on the positive f n integers such that is the nth term of the sequence for each n 1 2 3 K Definition A function f whose domain is the set of all positive integers 1 2 3 K natural f n numbers is called an infinite sequence The function value is called the nth term of the sequence Notation We denote sequences in any of the following forms a1 a2 a3 K an K an n 1 an a The subscript n that appears in n is called an index and it indicates the order of terms in the sequence The choice of a starting index is arbitrary but sequences usually begin with n 0 or n 1 1 4 7 10 13 16 K can be defined in two ways explicitly or implicitly First The sequence we have the rule that each term of the sequence is 3 more than the previous term that is a2 a1 3 a3 a2 3 a4 a3 3 and so forth In general we see that a1 1 and an 1 an 3 for n 1 2 3 K This way of defining a sequence is called a recurrence relation or an implicit formula It specifies the initial term of the sequence and gives a general rule for computing the next term of the sequence from previous terms Suppose you want to find a147 What should you do a 3n 2 The nth term is determined by n for n With this explicit formula the nth term of the sequence is determined directly from the value of n an n 1 Use the explicit formula for to write the first four terms of the sequence n 1 n an 2 n 1 1 What does the n do to the terms Consider the following sequence bn 3 6 12 24 48 K a Find the next two terms of the sequence b Find a recurrence relation that generates the sequence c Find an explicit formula for the nth term of the sequence Limit of a Sequence Perhaps the most important question about a sequence is this If you go farther and farther out in a K a10 000 K a100 000 K the sequence 100 how do the terms of the sequence behave Do they approach a specific number and if so what is that number Or do they grow in magnitude without bound Or do they wander around with or without a pattern The long term behavior of a sequence is described by its limit Definition Limit of a Sequence a If the terms of a sequence n approach a unique number L as n increases then we say lim an L n exists and the sequence converges to L If the terms of the sequence do not approach a single number as n increases the sequence has no limit and the sequence diverges Limits of sequences are really no different from limits at infinity of functions except that the variable n assumes only integer values as n Theorem 8 1 Limits of Sequences from Limits of Functions lim f x L f n an Suppose f is a function such that for all positive integers n If x then a the limit of the sequence n is also L Your textbook will sometimes convert the terms of a sequence to a function of x You can take the limit as n of the terms of the sequence directly Theorem 8 2 Properties of Limits of Sequences a b Assume that the sequences n and n have limits A and B respectively Then lim an bn A B 1 n lim can cA 2 n where c lim anbn AB 3 n a A lim n n b B provided B 0 n 4 Theorem 8 6 Growth Rates of Sequences The following sequences are ordered according to increasing growth rates as n that is if a lim n 0 an appears before bn in the list then n bn ln n n n q p p ln r n n p s b n n n n The ordering applies for positive real numbers p q r and s and b 1 I expect to see work Recall the limits of indeterminate form that may require use of l H pital s rule 0 0 00 1 and 0 Do the following sequences converge or diverge an n 0 8n 2 800n 5000 2 2n 1000n 2 n 0 bn n 1 23 n n 1 n 5 n cn n 1 n n 1 0 Determine the behavior of an ln n ln n 1 an and bn bn n 2 n n Terminology for Sequences a a an is said A sequence n in which each term is greater than or equal to its predecessor n 1 a to be nondecreasing A sequence n in which each term is less than or equal to its a an is said to be nonincreasing predecessor n 1 A sequence that is either nonincreasing or nondecreasing is said to be monotonic it progresses in only one direction an a M is bounded from above if there exists a number M such that n for all n In such a case the number M is said to be an upper bound for an If M is an upper a a bound for n but no number less than M is an upper bound for n then M is the least …