10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.

Transcription

1 10.1 Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1 Examples: EX1: Find a formula for the general term a n of the sequence, assuming the pattern of the first few terms continues. { 1/4, 2/9, 3/16, 4/25,...} EX2: List the first six terms of the sequence defined by a n = n 2n + 1 1

2 Limits and Convergence A sequence {a n } has the limit L and we write lim a n = L n if we can make the terms of a n as close to L as we like by taking n sufficiently large. If lim n a n exists, we say the sequence converges. Otherwise, we say the sequence diverges. Theorem: If lim x f(x) = L and f(n) = a n when n is an integer, then lim n a n = L. Example: Does the sequence in EX2 have a limit? If so, find it. Recall Limit Laws... Also, recall the Squeeze Theorem... Examples: Determine whether the sequence is convergent or divergent: EX3: a n = n 1 + n EX4: a n = ( 1)n n 3 n 2 + 2n EX5: a n = (cos n)n ln(n + 1) 2

3 Increasing sequence: Decreasing sequence: Montonic sequence: Bounded sequence: Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent. Example: Consider the sequence: {0.1,0.12,0.123,0.1234,..., , , ,... }. Show this sequence converges. EX: Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bonded? a n = 2n 3 3n + 4 3

4 10.2 Summing an Infinite Series Example: A series is an addition of the members of an infinite sequence. Terms vs. Partial Sums: Sequences and Sequences of Partial Sums: Convergent vs. Divergent Series: 4

5 Geometric Series is convergent if r and its sum is If r, the geometric series is divergent. Where does this formula come from? Examples: Determine whether the following converge or diverge: n=0 7 3 n 11 n 4 5 n 5 4 n 6 n 5

6 Theorem: If the series a n is convergent, then. This leads to... Divergence Test: If lim a n does not exist or if lim a n 0, then the series n n a n is divergent. Converse of above theorem? Note: The harmonic series Telescoping sums 1 n is divergent. Example: Determine whether the series is convergent or divergent. If it is convergent, find its sum. 3 n(n + 3) 6

7 Examples: Determine whether the following converge or diverge: ( 1) n n sin( n + 1 ) ( 1) 2n [ ] 3 n(n + 3) ( 1 2 )n 7

8 10.3 Convergence of Series with Positive Terms Integral Test Integral Test Suppose f is a,, function on [1, ) and let a n = f(n). Then the series a n is convergent if and only if the improper integral 1 f(x) dx is convergent. Example: n=2 1 n(ln n) 2 For what values of p is the series 1 n p convergent? 8

9 The p-series is convergent if and divergent if. Testing by Comparing Comparison Test Assume that there exists M > 0 such that 0 a n b n for n M (a) If b n is convergent then a n is also convergent. (b) If a n is divergent then b n is divergent. What do I compare to? **Note: Examples: Determine whether the series converges or diverges: n=3 1 n 2 5 Limit Comparison Test: Suppose that a n and b n are series with positive terms. If a n lim = c where c is a finite number, then n b n If c > 0, then either both seires converge or both diverge. If c = 0 and b n converges, then a n converges. 9

10 Examples: Determine whether the series converge or diverge. n 2 + 2n + 3 3n 4 + 7n n n n + 17 n 2 ln n n 1 10

11 10.4 Absolute and Conditional Convergence We ve been dealing with series that have positive terms. positive and negative terms? What about those series with A series is absolutely convergent if the series of absolute values a n is convergent. Theorem: If a series a n is absolutely convergent, then it is convergent. An infinite series a n is conditionally convergent if it converges, but a n diverges. EX1: Verify the convergence of S = What about a series that is not absolutely convergent? Example: Leibniz Test for Alternating Series (AST): If the alternating series S = ( 1) n 1 b n = b 1 b 2 + b 3 b (b n > 0) satisfies (i) b n+1 b n for all n and (ii) lim n b n = 0 then the series is convergent. Furthermore, 0 S a 1 and S 2N S S 2N+1 for all N. 11

12 Determine if the series converges absolutely, conditionally, or not at all: ( 1) n+1 n (1.1) n cos nπ n=0 Estimating a series: If S = ( 1) n 1 a n is the sum of an alternating series that is convergent by the AST, then S N S a n+1. EX: Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? ( 1) n+1 n 6 ( error < ) 12

13 10.5 The Ratio and Root Test The Ratio Test: (i) If lim a n+1 n a n = L < 1, then (ii)if lim a n+1 n a n = L > 1, then (iii) If lim a n+1 n a n = 1, then Example: Determine if the series converges or diverges: 5 n n n ( 1) n n! 100 n Check if b n converges where {b n } is the recursive sequence b 1 = 1, b n+1 = (1 + 1 n )n b n 13

14 The Root Test: Let {a n } be a sequence and assume that the following limit exists: L = lim n n an (i) If L < 1, then (ii) If L > 1, then (iii) If L = 1, then Example: Determine whether ( ) n n converges. 2n

15 10.6 Power Series A power series centered at the point x = c is an infinite seires of the form F (x) = a n (x c) n = a 0 + a 1 (x c) + a 2 (x c) 2 + a 3 (x c) n=0 **Note: The value of a power series depends on the value of x. Radius of Convergence: Let F (x) = n=0 a n(x c) n. There are three cases for the value of R, the radius of convergence of F (x): (i) F (x) converges only for x = c and R = 0, or (ii) F (x) converges for all x and R =, or (iii) There is a number R > 0 such that F (x) converges absolutely if x c < R and diverges if x c > R. It may or may not converge at the endpoints x c = R. Intervals of convergence: Geometric Series Example: Find the radius and interval of convergence for x n. n=0 15

16 Compute the radius and interval of convergence of: x 2n (2n)! (x 1) n n ( 2) n n (x + 3) n Show that the p-power series x n for p > 1 have [ 1, 1] as their interval of convergence. np 16

17 New viewpoint: **Note: All polynomials are their own representations as power series. Recall: 1 1 x = x n. n=0 Derive power series for the following functions: a) x (b) x x + 1 (c) x 2 Theorem: If the power series a n (x c) n has radius of convergence R > 0, then the function defined by f(x) = a n (x c) n n=0 is differentiable on the interval (c R, c + R) and (i) f (x) = (ii) f(x) dx = The radii of convergence of both of these power series is also R. **Note: Substituting a possibly a different radius of convergence. will yield a new power series with 17

18 1 Example: Derive the power series representation for (1 + x) Find a power series representation for the function f(x) = arctan(x/3) and determine the radius of convergence. Approximation vs. Equality: 28. Use a power series to approximate the definite integral to six decimal places x x 4 18

19 8.4 Taylor Polynomials How do we approximate a nonpolynomial function with a polynomial? The nth Taylor polynomial for f centered at c for n = 1,..., k is T n (x) = f(x) + f (x) 1! (x c) + f (x) 2! When c = 0, T n (x) is called the. (x c) f (n) (x) (x c) n. n! Let s look at the nth Maclaurin polynomials for some familiar functions: a) f(x) = e x b) g(x) = sin x c) h(x) = ln x Ex: Find T n (x) at centered at x = 4 for all n with f(x) = 1 x 1. For a given function f(x), the nth remainder for f(x) at x = c is given by R n (x) = f(x) T n (x). Note: The error is just the absolute value of R n (x) and f(x) = T n (x) + R n (x). Taylor s Theorem: Assume f (n+1) (x) exists and is continuous. Then, R n (x) = 1 n! x c (x u) n f (n+1) (u) du. We won t use Taylor s Theorem directly. Instead, we ll use it to estimate the error. 19

20 Error Bound: Assume that f (n+1) (x) exists and is continuous. Let K be a number such that f (n+1) (u) K for all u between c and x. Then, T n (x) f(x) K x c n+1. (n + 1)! Example: Find n such that T n (1) e 10 6, where T n (x) is the Maclaurin polynomial for f(x) = e x. 20

21 10.7 Taylor Series Theorem: If f has a power series representation at c, that is, if f(x) = a n (x c) n n=0 x c < R then its coefficients are given by a n = f (n) (c). n! The series in the above theorem is known as the Taylor series of the function f at c. When c = 0, this is known as the Maclaurin series. Find the Maclaurin series for f(x) = e x and find its radius of convergence. Find the Maclaurin series for f(x) = sin x using the definition of a Maclaurin series: 21

22 Theorem: Let f(x) be an infinitely differentiable function on the open interval I = (c R, c + R) with R > 0. Assume there exists K 0 such that for all k 0, f (k) (x) K for all x in I. Then f(x) is represented by its Taylor series in I. Proof: Example: Show that the Maclaurin series found for sin x is valid for all x. Important Maclaurin series: 1 1 x = e x = sin x = cos x = tan 1 x = 22

23 Examples: Derive the power series for the following: sin x 2 sin 5x e x3 Ex: Evaluate the integral as an infinite series: sin x x dx Ex: Find the sum of the series 1 ln 2 + (ln 2)2 (ln 2)3 2! 3!