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Chapter 11 Highlight part I

Chapter 11 Highlight part I - Chapter 11 H ighlights...

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Chapter 11 Highlights Section 11.1: Sequences A sequence is an infinite list of numbers , , ,…, ,… a1 a2 a3 an , where an is the nth term of the sequence. != - - … ( ) n nn 1n 2 2 1 (e.g. != = 4 4∙3∙2∙1 24 or != = 5 5∙4∙3∙2∙1 120 ). By convention != 0 1 . →∞ = limn an L means that, as you plug bigger and bigger values in for n , an gets closer and closer to L . Theorem (related function) : If ( )→ f x L as →∞ x and = fn an , then an L as →∞ n . This basically means that you can replace the n in fn with an x and take the limit as x goes to infinity instead to find the limit of the sequence. This allows you to do things like apply L’Hopital’s Rule if necessary. A sequence is monotonically increasing if each terms is smaller than the next and is monotonically decreasing if each terms is bigger than the next. A sequence is bounded above if there is some number bigger than every term in the sequence and is bounded below if there is some number smaller than every term in the sequence. A series is bounded if it’s bounded above and below.
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In using Principle of Mathematical Induction , we follow these steps: Step 1 : Prove the statement pn is true when = n 1 . Step 2 : Assume that pn is true for = n k , and deduce (prove) that pn is true for = + n k 1 . This then allows you to say that pn is true for all positive integer n . Mathematical Induction is a way to determine whether a recursive sequence is convergent. Section 11.2: Series A series is an infinite sum = ∞ = + + +… n 1 an a1 a2 a3 . The nth partial sum of a series is = + + +…+ Sn a1 a2 a3 an (the sum of the first n terms). The remainder = + + + + + +… Rn an 1 an 2 an 3 is what’s left over. If →∞ = limn Sn S exists, then we denote this limit by = = ∞ = + + +… S n 1 an a1 a2 a3 . So + + +… = + + +…+ + + + + + + +… a1 a2 a3 S a1 a2 a3 anSn an 1 an 2 an 3 Rn A series converges if and only if the sequence of the partial sums converges.(in turn, the remainder goes to 0).
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Informally, the series converges if the infinite sum actually adds up to some real number. Otherwise, it diverges. A geometric series is a series of the form + + + a ar ar2 ar3 which is equal to - a1 r provided that < r 1 . The series diverges if r 1 . Theorem (Test for divergence) : If an↛0 (the an ’s don’t go to 0 ) , then the series diverges. Telescoping Series : This special kind of series has the characteristics that the middle terms of the nth partial sum collapses like an old telescope (you’ve seen in the Pirate’s of the Caribbean).
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