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

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1 8. Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = Examples: 6. Find a formula for the general term a n of the sequence, assuming the pattern of the first few terms continues. { /4, 2/9, 3/6, 4/25,...} 3. List the first six terms of the sequence defined by a n = n 2n +

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 # 3 have a limit? If so, find it. Recall Limit Laws... Also, recall the Squeeze Theorem... Examples: Determine whether the sequence is convergent or divergent: 4. a n = n + n 6. a n = ( )n n 3 n 2 + 2n 2 + a n = (cos n)n ln(n + ) 2

3 Increasing sequence: Decreasing sequence: Montonic sequence: Bounded sequence: Monotonic Sequence Theorem: Every bounded, monotonic sequence is convergent. Example: Consider the sequence: {0.,0.2,0.23,0.234,..., , , ,... }. Show this sequence converges. 42. Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bonded? a n = 2n 3 3n + 4 3

4 8.2 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. Examples: Determine whether the following converge or diverge: k n, k > 4 5 n 5 4 n 6 n Theorem: If the series a n is convergent, then. This leads to... Test for Divergence: If lim n a n does not exist or if lim n a n a n is divergent. 0, then the series Converse of above theorem? 5

6 Note: The harmonic series Telescoping sums n is divergent. Example: Determine whether the series is convergent or divergent. If it is convergent, find its sum. 3 n(n + 3) Examples: Determine whether the following converge or diverge: ( ) n n sin( n + ) 6

7 ( ) 2n [ ] 3 n(n + 3) ( 2 )n Example: Write 0.42 as a ratio of integers. 7

8 8.3 The Integral and Comparison Tests; Estimating Sums Integral Test Integral Test Suppose f is a,, function on [, ) and let a n = f(n). Then the series a n is convergent if and only if the improper integral f(x) dx is convergent. Example: n=2 n(ln n) 2 For what values of p is the series n p convergent? 8

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

10 Examples: Determine whether the series converge or diverge. n 2 + 2n + 3 3n 4 + 7n 3 + n 2 + 3n + 7 2n + 7 n 2 ln n n Estimating the sum of a series Remainder Estimate for Integral Test: Suppose f(k) = a k, where f is continuous, positive, decreasing function for x n and a n is convergent. If R n = s s n, then n+ f(x) dx R n n f(x) dx 0

11 Example: Using the series n 2, find values of n for which the remainder R n < 0.0, and then values of n for which R n < Then do the same for n 4. Remainder estimate also gives bounds for the sum, s:

12 8.4 Other Convergence Tests Examples: The Alternating Series Test (AST): If the alternating series ( ) n b n = b b 2 + b 3 b (b n > 0 satisfies (i) b n+ b n for all n and (ii) lim n b n = 0 then the series is convergent. Determine the convergence of the following series: ( ) n+ n (.) n cos nπ n=0 Alternating Series Estimation Theorem: If s = ( ) n b n is the sum of an alternating series that is convergent by the AST, then R n = s s n b n+. 2

13 3. 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? ( ) n+ n 6 ( error < ) A series is absolutely convergent if the series of absolute values a n is convergent. Examples: ( ) n+ n (.) n cos nπ n=0 Theorem: If a series a n is absolutely convergent, then it is convergent. The Ratio Test: (i) If lim a n+ n a n = L <, then (ii)if lim a n+ n a n = L >, then (iii) If lim a n+ n a n =, then 3

14 Example: Determine if the series converges or diverges: 5 n n n ( ) n n! 00 n Check if b n converges where {b n } is the recursive sequence b =, b n+ = ( + n )n b n The Root Test: Let {a n } be a sequence and assume that the following limit exists: L = lim n n an (i) If L <, then (ii) If L >, then (iii) If L =, then Example: Determine whether ( ) n n converges. 2n + 3 4