Lecture 23 : Sequences A Sequence is a list of numbers written in order. fa1; a2; a3;::: g The sequence may be infinite. The n th term of the sequence is the n th number on the list. On the list above a1 = 1st term, a2 = 2 nd term, a3 = 3 rd term, etc.... th Example In the sequence f1; 2; 3; 4; 5; 6;::: g, we have a1 = 1; a2 = 2;::: . The n term is given by an = n. Some sequences have patterns, some do not. Example If I roll a 20 sided die repeatedly, I generate a sequence of numbers, which have no pattern. Example The sequences f1; 2; 3; 4; 5; 6;::: g and f1; −1; 1; −1; 1;::: g have patterns. Sometimes we can give a formula for the n th term of a sequence, an = f(n). Example For the sequence f1; 2; 3; 4; 5; 6;::: g; we can give a formula for the n th term. an = n. Example Assuming the following sequences follow the pattern shown, give a formula for the n-th term: f1; −1; 1; −1; 1;::: g {−1=2; 1=3; −1=4; 1=5; −1=6;::: g Factorials are commonly used in sequences 0! = 1; 1! = 1; 2! = 2 · 1; 3! = 3 · 2 · 1; : : : ; n! = n · (n − 1) · (n − 2) ····· 1: Example Find a formula for the n th term in the following sequence ( ) 2 4 8 16 32 ; ; ; ; ; : : : ; a = ; 1 2 6 24 120 n Below we show 3 different ways to represent a sequence: n1 2 3 n o n n o1 n ; ; ;:::; ;::: an = : 2 3 4 n + 1 n + 1 n=1 n + 1 1 n−3 5 −7 n (2n + 1) o n n (2n + 1)o n (2n + 1) ; ; ;:::; (−1) ;::: (−1) an = (−1) : 3 9 27 3n 3n n=1 3n 1 ne e2 e3 en o nen o1 en ; ; ;:::; ;::: an = : 1 2 6 n! n! n=1 n! Graph of a Sequence A sequence is a function from the positive integers to the real numbers, with f(n) = an. We can draw a graph of this function as a set of points in the plane. The points on the graph are (1; a1); (2; a2); (3; a3);:::; (n; an);::: 1 (−1)n 1 n 2n3−1 o Example Below, we show the graphs of the sequences f n gn=1 and n3 . n=1 3 H-1Ln 2 n - 1 points Hn, L, n = 1...100 points Hn, L, n = 1...100 n n3 2.0000 0.10 1.9999 0.05 1.9998 20 40 60 80 100 1.9997 -0.05 1.9996 -0.10 20 40 60 80 100 We can see from these pictures that the graphs get closer to a horizontal asymptote as n ! 1, y = 0 for the sequence on the left and y = 2 for the sequence on the right. Algebraically this means that as (−1)n 2n3−1 n ! 1, we have n ! 0 and n3 ! 2. Limit of a Sequence Definition A sequence fang has limit L if we can make the terms an as close as we like to L by taking n sufficiently large. We denote this by lim an = L or an ! L as n ! 1: n!1 If limn!1 an exists (is finite), we say the sequence converges or is convergent. Otherwise, we say the sequence diverges. 1 Graphically: If limn!1 an = L, the graph of the sequence fangn=1 has a unique horizontal asymptote y = L. Equivalent Definition A sequence fang has limit L and we write lim an = L or an ! L as n ! 1 n!1 if for every > 0 there is and integer N with the property that if n > N then jan − Lj < . 2 Determining if a sequence is convergent. Using our previous knowledge of limits : Theorem If limx!1 f(x) = L and f(n) = an, where n is an integer, then limn!1 an = L. Example Determine if the following sequences converge or diverge: n2n − 1o1 n2n3 − 1o1 ; 2n n=1 n3 n=1 We can use L'Hospital's rule to determine the limit of f(x) if we have an indeterminate form. Example Is the following sequence convergent? n n o1 2n n=1 Diverging to 1. limn!1 an = 1 means that for every positive number M, there is an integer N with the property if n > N; then an > M: In this case we say the sequence fang diverges to infinity. Note: If limx!1 f(x) = 1 and f(n) = an, where n is an integer, then limn!1 an = 1. n 1 Example Show that the sequence fr gn=1, r ≥ 0, converges if 0 ≤ r ≤ 1 and diverges to infinity if r > 1. The usual Rules of Limits apply: If fang and fbng are convergent sequences and c is any constant then lim can = c lim an n!1 n!1 lim (an + bn) = lim an + lim bn n!1 n!1 n!1 lim (anbn) = lim an · lim bn n!1 n!1 n!1 lim (an − bn) = lim an − lim bn n!1 n!1 n!1 an limn!1 an lim c = c lim = if lim bn 6= 0 n!1 n!1 bn limn!1 bn n!1 3 p p h i lim an = lim an if p > 0 and an > 0 n!1 n!1 In fact if limn!1 an = L and f(x) is a continuous function at L, then lim f(an) = f(L): n!1 Example Determine if the following sequence converges or diverges and if it converges find the limit. r n 2n + 1 1 o1 3 − : n n n=1 Note We cannot always find a function f(x) with f(n) = an. The Squeeze Theorem or Sandwich Theorem can also be applied : Ifan ≤ bn ≤ cn for n ≥ n0 and lim an = lim cn = L; then lim bn = L: n!1 n!1 n!1 Example Find the limit of the following sequence n2n o1 ; n! n=1 Alternating Sequences For any sequence, we have −|anj ≤ an ≤ janj. We can use the squeeze theorem to see that if lim janj = 0; then lim an = 0: n!1 n!1 In fact any sequence with infinitely many positive and negative values converges if and only if limn!1 janj = 0 n 0 0 Let fang = f(−1) ang where an > 0 0 n 0 • If limn!1 an = L 6= 0, then limn!1(−1) an does not exist. 0 n 0 • If limn!1 an = 1, then limn!1(−1) an does not exist. 0 n 0 • If limn!1 an does not exist, then limn!1(−1) an does not exist. 4 0 Below, we show a picture of a sequence where, as in the first case above, limn!1 an = L 6= 0. H-1Ln H2 n + 5L points Hn, L, n = 1...100 n 4 2 20 40 60 80 100 -2 -4 -6 n 0 0 Theorem If fang is an alternating sequence of the form (−1) an where an > 0 , then the al- ternating sequence converges if and only if limn!1 janj = 0 or (for the sequence described above) 0 limn!1 an ! 0. n+1 0 (also true for sequences of form (−1) an or any sequence with infinitely many positive and negative terms) Example Determine if the following sequences converge: n 2n + 1o1 n 2n + 1o1 (−1)n ; (−1)n n2 n=1 n n=1 5 Monotone Sequences Definition A sequence fang is called increasing if an < an+1 for all n ≥ 1, or a1 < a2 < a3 < : : : : A sequence fang is called decreasing if an > an+1 for all n ≥ 1, or a1 > a2 > a3 > : : : : A sequence fang is called monotonic if it is either increasing or decreasing. Definition A sequence fang is bounded above if there is a number M for which an ≤ M for all n ≥ 1: A sequence fang is bounded below if there is a number m for which an ≥ m for all n ≥ 1: A sequence that is bounded above and below is called Bounded. Theorem Every bounded monotonic sequence is convergent. (This theorem will be very useful later in determining if series are convergent.) To check for monotonicity 0 If we have a differentiable function f(x) with f(n) = an, then the sequence fang is increasing if f (x) > o 0 and the sequence fang is decreasing if f (x) < o. Example Show that the following sequence is monotone and bounded and hence converges. −1 1 ftan (n)gn=1 6.