Math 123- Shields Infinite Sequences Week 3 3.2 Introduction By a sequence we loosely mean an ordered list of numbers, 2; 4; 6; 8; 10; 12: We refer to entries of the sequence as the terms of the sequence and to each term in the sequence we assign a positive whole number called its index. The index refers to the term's place within the sequence. For example the above sequence could be written a1; a2; a3; a4; a5; a6 where a1 = 2; a2 = 4; etc. The indices must be sequential, however, they need not start at 1. We could have instead notated the sequence by , b10; b11; b12; b13; b14; b15 where b10 = 2; b11 = 4; etc. Sequences may also be infinite in length. In fact we will only deal with infinite sequences in this class so I will often drop the adjective when I speak. As an example of an infinite sequence we could imagine a sequence which extends the above pattern on towards infinity. 2; 4; 6; 8; 10; 12;::: The word pattern is problematic and must be made more rigorous. Sometimes people colloquially speak of a "last term" in an infinite sequence. This by itself is nonsense, but it can be given some real sort of meaning as we will see. 3.3 Infinite Sequences & Convergence Notation: Let N denote the collection of non-negative whole numbers: 0; 1; 2;::: and N≥m the collection of non-negative whole numbers not smaller than m. For example N≥4 denotes the numbers 4; 5; 6;:::. Definition A sequence of real numbers is a function f whose domain is N≥m and range is the real numbers. This agrees with the intuitive idea of a sequence discussed in the introduction by taking an = f(n). For example, the sequence talked about there could be described by the function f(n) = 2n with domain N≥1. Although the formal definition is a function and it can be helpful to remember this, sometimes it is unnec- essary and we instead operate with the informal definition of a sequence as a list of numbers. k Notation The finite sequence am; am+1; am+2; : : : ; ak is denoted by fangn=m and the infinite sequence am; am+1; am+2;::: by fangn=m. A sequence may defined in any number of ways, but the salient point is that it must be done so unambigu- ously so that all parties agree on precisely what the terms of the sequence are. In this class we mainly do so in one of two ways. 1 Math 123- Shields Infinite Sequences Week 3 1) As an Explicit Function of n The most common way we will define a sequence is by giving an explicit function which allows us to compute the nth term of the sequence by simply evaluating the function. Examples: 1 1 1 (i) f n gn=1 = 1; 2 ; 3 ;::: 1 From this we can easily compute any term directly. For example a100 = 100 . 1−n −1 −2 (ii) fe gn=0 = e; 1; e ; e ;::: n+1 n(−1) o 1 1 1 (iii) = 1; − 4 ; 9 ; − 16 ;::: n2 n=1 n no (iv) an = 2 = 1; 2; 4; 8; 16; 32;::: n=0 2) Recursively Occasionally it is more convenient to describe a sequence by giving some necessary initial values of the sequence and a formula to compute new values from the previous terms. Examples: (i) a0 = 1; a1 = 1; an = an−1 + an−2 Using this recursive formula we can compute as many terms of the sequence as is necessary. For example a2 = a2−1 + a2−2 = a1 + a0 = 1 + 1 = 2 and a3 = a3−1 + a3−2 = a2 + a1 = 2 + 1 = 3 The sequence described is the well known Fibonacci Sequence. Note that with this method of defining a sequence we cannot immediately compute an arbitrary value such as a100. (ii) a0 = 1; an = 2an−1. Here a1 = 2a0 = 2 · 1 = 2; a2 = 2a1 = 2 · 2 = 4; a3 = 2a2 = 2 · 4 = 8. Notice this gives the same n sequence as the one described by the formula an = 2 . 9 (iii) a = 0; a = a + . 0 n n−1 10n This describes the sequence 0; 0:9; 0:99; 0:999; 0:9999; 0:99999;::: A primary concern of ours will be the progression of sequences and whether their terms begin to settle down around a value or not. n o Definition A sequence an is said to converge if lim an exists and is finite. Otherwise it is said to diverge. n!1 If lim an = L we will say that the sequence converges to L. n!1 Real Definition which you can ignore but can also be useful if you take the time to understand n o it A sequence an is said to converge to L if for any real number > 0 there is some non-negative whole number N such that for all n ≥ N we have that an − L < . 2 Math 123- Shields Infinite Sequences Week 3 Convergence sequence are nice in that they obey the expected arithmetic laws. Theorem Let fang; fbng be sequences converging to a and b respectively and c a constant. (i) fan ± bng converges to a ± b. (ii) fc · ang converges to c · a. (iii) fan · bng converges to a · b. an a (iv) f g converges to b if b 6= 0. bn If we have a sequence given explicitly given as a function of n, then our primary technique to determine if a sequence converges or diverges will be to treat the sequence as a real function and use the techniques developed in Calc I to analyze it. This is justified by the following theorem. Theorem Let fang be a sequence and f(x) a function such that f(n) = an. Then if lim f(x) = L, then x!1 lim an = L. n!1 Examples: n 1 o 1 (a) n converges to 0 since f(x) = x agrees with the sequence at all positive integer values and n=1 1 lim = 0 x!1 x n o (b) n diverges since f(x) = x agrees with the sequence at all positive integer values and n=1 lim x = 1 x!1 n n o x (c) n+1 converges to 1. Take f(x) = x+1 . Then, n=0 x x 1 1 1 lim = lim · = lim = = 1 x!1 x!1 1 x!1 1 x + 1 x 1 + x 1 + x 1 + 0 n n o x (d) en converges to 0. Take f(x) = ex . Then, n=0 x 1 lim = x!1 ex 1 By L'Hopital's Rule this limit is equivalent to 1 lim = 0 x!1 ex Note: Be careful. The theorem above only says what is says. It does not say that if lim f(x) does not x!1 exists, the lim an does not exist. For example take an = sin(nπ). Then an = 0 for all n so the sequence x!1 must converge to zero. However, taking f(x) = sin(nx) gives that lim f(x) does not exist as it oscillates x!1 between −1 and 1. The examples show the previous theorem to be a powerful tool in analyzing the convergence of sequences. Of course, it alone will be insufficient to handle all sequences. Some other potentially useful theorems are documented here. 3 Math 123- Shields Infinite Sequences Week 3 n o Theorem Let an be a sequence. n o (i) If lim an = 0, then an converges to 0. n!1 n o (ii) If lim a2n = L and lim a2n+1 = L, then an converges to L. n!1 n!1 n o (iii) If lim a2n = L and lim a2n+1 = M where L 6= M, then an diverges. n!1 n!1 n o (iv) If bn ≤ an ≤ cn and lim bn = lim cn = L, then an converges to L. n!1 n!1 n o n o (v) If an converges to L, then any subsequence of an converges to L. n o (vi) If an converges to L and f(x) is continuous at L, then lim f(an) = f(L). n!1 Examples: n(−1)n+1 o (i) converges to 0 since n n=1 (−1)n+1 1 lim = lim = 0 n!1 n n!1 n n n+1 1 o (ii) π + (−1) n converges to π since n=1 2n+1 1 1 lim a2n = lim π + (−1) = lim π − = π − 0 = π n!1 n!1 n n!1 n and 2n+2 1 1 lim a2n+1 = lim π + (−1) = lim π + = π + 0 = π n!1 n!1 n n!1 n n 1 − no (iii) (−1)n diverges since n n=1 2n 1 − n 1 − n lim a2n = lim (−1) = lim = −1 n!1 n!1 n n!1 n and 2n+1 1 − n 1 − n lim a2n+1 = lim (−1) = lim − = 1 n!1 n!1 n n!1 n nsin(n)o −1 sin(n) 1 (iv) converges to 0 since ≤ ≤ and n n=1 n n n ±1 lim = 0 n!1 n 1 (v) Assume that the sequence given by a0 = 0:1; an = 1 + converges to L. Knowing this we may 1 + an−1 determine L. p 1 2 2 lim an = lim =) L = 1 + =) L + L = 1 + L + 1 =) L = 2 =) L = ± 2 n!1 n!1 1 + L p It does not take much more work to show that L = 2 since we begin with a positive number and can therefore only return positive numbers based on the recursive formula.