Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a . n 1 − x 1 1 1 Example 1: Is 2 + 1 + + + + ::: geometric? 2 4 8 1 Yes 2 No Sequences and series Calculus and Differential Equations II Geometric series Example 2: Is 1 + x + 2x2 + 3x3 + 4x4 + ::: geometric? 1 Yes 2 No Example 3: Find 3(0:1) + 3(0:1)2 + 3(0:1)3 + ··· + 3(0:1)10. n X Example 4: Assume uk is geometric and that uk 6= 0 for k=1 n X 1 all k's. Is geometric as well? uk k=1 1 Yes 2 No Sequences and series Calculus and Differential Equations II Convergence of series n X Recall that we defined the partial sum Sn as Sn = uk . k=1 If Sn converges and lim Sn = S, then one says that the n!1 1 X infinite series uk converges, and that its sum is equal to S. k=1 One writes 1 X uk = S: k=1 If lim Sn does not exist or is infinite, one says that the series n!1 1 X uk diverges. k=1 Sequences and series Calculus and Differential Equations II Convergence of geometric series Recall that an infinite geometric series is written as an infinite sum of the form a + ax + ax2 + ··· + axn + ::: a; x 6= 0 a If jxj < 1, the above series converges to . 1 − x If jxj > 1, the series diverges. n X If x = 1, a = na and the series diverges. k=1 n X If x = −1, a = a − a + a − a + ::: , and the series diverges. k=1 Sequences and series Calculus and Differential Equations II More on convergence of series n n X X If an and bn converge, and if k is a constant, then n=1 n=1 n 1 1 X X X 1 (an + bn) converges to an + bn n=1 n=1 n=1 n 1 X X 2 k an converges to k an. n=1 n=1 Changing a finite number of terms in a series does not change its convergence properties. 1 X If lim an is not zero, or if it does not exist, then an n!1 n=1 diverges. 1 1 X X If an diverges, so does k an, for k 6= 0. n=1 n=1 Sequences and series Calculus and Differential Equations II Comparison of series with integrals There is a direct correspondence between convergence of series and convergence of integrals. This is called the integral convergence test. Assume that an = f (n), where f is a continuous, decreasing function that remains positive for x > b. Then, n Z 1 X If f (x) dx converges, so does an. b n=1 n Z 1 X If f (x) dx diverges, so does an. b n=1 This can be understood by writing the series as a left or right sum approximation of the improper integral. Sequences and series Calculus and Differential Equations II Comparison of series with integrals (continued) 1 X 1 Example 1: Does converge? en n=1 1 Yes 2 No 1 X 4 Example 2: Does converge? (2n + 3)3 n=1 1 Yes 2 No 1 X 1 Example 3: Does converge? 1 + n n=1 1 Yes 2 No Sequences and series Calculus and Differential Equations II Tests for convergence Theorem 1: The Comparison Test Suppose 0 ≤ an ≤ bn for all n's. Then, 1 1 X X 1 If bn converges, so does an. n=1 n=1 1 1 X X 2 If an diverges, so does bn. n=1 n=1 Theorem 2: Suppose an > 0 and bn > 0 for all n's. If a lim n = c; c > 0; n!1 bn 1 1 X X then the two series an and bn either both converge or n=1 n=1 both diverge. Sequences and series Calculus and Differential Equations II Tests for convergence (continued) Definitions: 1 1 X X 1 The series an is absolutely convergent if janj converges. n=1 n=1 1 1 X X 2 The series an is conditionally convergent if janj diverges n=1 n=1 1 X but an converges. n=1 Theorem 3: Absolute convergence implies convergence 1 1 X X If janj converges, so does an. n=1 n=1 Sequences and series Calculus and Differential Equations II Tests for convergence (continued) Theorem 4: The Ratio Test 1 X Consider the series an and suppose that n=1 an+1 lim = L: n!1 an Then, 1 X 1 If L < 1, then the series an converges. n=1 1 X 2 If L > 1, then the series an diverges. n=1 3 If L = 1, more information is needed to conclude. Sequences and series Calculus and Differential Equations II Tests for convergence (continued) Theorem 5: The Alternating Series Test 1 X n−1 Consider the series (−1) an and suppose that n=1 lim an = 0; and 0 < an+1 < an: n!1 1 X n−1 Then, the alternating series (−1) an converges. n=1 Theorem 6: Error bounds for alternating series Assume that the conditions of Theorem 5 are satisfied, and let 1 n X n−1 X k−1 S = (−1) an; Sn = (−1) ak : n=1 k=1 Then, jS − Snj < an+1 for all n's. Sequences and series Calculus and Differential Equations II Example 1 X (−1)n−1 Consider the series . n n=1 Can we use the ratio test to conclude convergence? 1 Yes 2 No Can we use the alternating series test instead? 1 Yes 2 No The series is 1 Conditionally convergent 2 Absolutely convergent How many terms should we include in the partial sum Sn to guarantee that Sn is an approximation of S correct to 3 decimal places? Sequences and series Calculus and Differential Equations II More examples 1 X 2 Does the series p converge? 2 + n n=0 1 Yes 2 No 1 X n + 2n Does the series converge? n 2n n=1 1 Yes 2 No 1 X n 2n Does the series converge? 3n n=5 1 Yes 2 No 1 X (−1)n−1 2n Does the series converge? n2 n=1 1 Yes 2 No Sequences and series Calculus and Differential Equations II Power series A power series is a series of the form 1 X n 2 n cn(x−a) = c0+c1 (x−a)+c2 (x−a) +···+cn (x−a) +::: n=0 Abel's lemma: If the power series converges for jx − aj = R0, then it converges absolutely for all x's such that jx − aj < R0. As a consequence, one can define its radius of convergence R: If the series only converges for x = a, one says that R = 0. If the series converges for all x's, one says that R = 1. Otherwise, R is the largest number such that the series converges for jx − aj < R. The interval of convergence of the power series is (a − R; a + R) plus any end point where the series converges. Sequences and series Calculus and Differential Equations II Power series (continued) Example 1: What are the radius of convergence and the 1 X (x − 2)n interval of convergence of the series ? n n=1 To find the radius of convergence of a power series, use the ratio test. Example 2: Find the interval of convergence of the power 1 X x2n series (−1)n . (2n)! n=0 Example 3: Find the interval of convergence of the power 1 X (x − 3)n series (−1)n . 2n n2 n=1 Sequences and series Calculus and Differential Equations II Review Which of the following sequences are monotone and bounded? 1 10n + 1 I : s = 10 − II : s = III : s = cos(n) IV : s = ln(n) n n n n n n 1 I only 3 II and IV 2 I and II 4 I, II, and III Which of the following sequences converge? cos(n) 1 sin(1=n) I : s = II : s = cos III : s = n e−n IV : s = n n n n n n 1=n 1 I and III 3 II and IV 2 I, II, and III 4 All of them Sequences and series Calculus and Differential Equations II Review (continued) Which of the following geometric series converge? I : 20 − 10 + 5 − 2:5 + ::: II : 1 − 1:1 + 1:21 − 1:331 + ::: III : 1 + 1:1 + 1:21 + 1:331 + ::: IV : 1 + y 2 + y 4 + y 6 ::: − 1 < y < 1 1 I only 4 II and IV 2 IV only 5 None of them 3 I and IV Sequences and series Calculus and Differential Equations II Review (continued) Which of the following statements are true? 1 X (a) If lim an = 0, then an converges.