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# PROBABILITY STATS Anthro 10

UCI

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This 3 page Class Notes was uploaded by Bryon Mueller on Saturday September 12, 2015. The Class Notes belongs to Anthro 10 at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/201908/anthro-10-university-of-california-irvine in anthropology, evolution, sphr at University of California - Irvine.

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Date Created: 09/12/15

101 Sequence and Series Sequences A sequence is a function with domain a set of successive integers For example consider a function f fn2n71 where the domain of f is the set of natural numbers N Usually the range value is symbolized more compactly with a symbol such as an Thus above example can be rewritten as an2n71 The domain is understood to be the set of natural numbers N unless stated to the contrary or the context indicates otherwise The elements in the range are called terms of the sequence a1 is the rst term a2 the second term and an the nth term or the general term a1 21 7 1 1 a2 22 7 1 3 a3 23 7 1 5 A sequence can be represented in form of the ordered list 11012 7 or in abbreviated form an For example 135 2n71 and 2 n 71 represent the same sequence that given by the function fn 2 n 7 1 If the domain of a function is a nite set of successive integers then the sequence is called a nite sequence If the domain is an in nite set of successive integers then the sequence is called an in nite sequence For example 13 57 9 is a nite sequence and 135 2n71 is an in nite sequence Some sequences are speci ed by a recursion formula that is a formula that de nes each term in terms of one or more preceding terms For example the famous sequence Fibonacci sequence is de ned as a1 17012 17an M71 M727 71 Z 3 1 2 The rst six terms of the Fibonacci sequence are given as following 04 1 a2 1 a3 a2a1112 a4 a3a2213 a5 a4a3325 a6 a5a4538 Series lf 11012 13 an is a sequence then the expression a1a2a3an is called a series lfthe sequence is nite the corresponding series is a nite series If the sequence is in nite the corresponding series is an in nite series For example 1 2 4 8 16 is a nite series and is an in nite series Series are often represented in a compact form called summation nota tion using the symbol For example 4 22k 2021222324 w Mal ak a1a2a3a4a5 w H H g k CICZCS Cn H Jquot H MA MA MA H MW 2 w H H The terms on the right are obtained from the expression on the left by successively replacing the summing index k with integers staring with the rst number indicated below 2 and ending with the number that appears above Example Write following expression without summation notation Solution 5717171 271 371 471 571 T TTTTT 0lg 2 3 4 5 D Example Write the following series using summation notation with the summing index start at k 1 1 1 1 1 1 1 7 7 7 7 7 7 7 7 1 2 3 4 5 6 Solution 1 is the sum of the sequence 1 1 1 1 1 7 27 37 47 57 6 Each of these terms can be written as the reciprocal of an integer and a power of 71 1 lHU lH gtlelHlDlH H H 71 671 7 lt gt lt6 Thus7 the series 1 can be represented as lt71 k 6 w H H D Power of 71 provide the alternation of sign If the terms of a series are alternately positive and negative it is called an altemating series Applications of sequence and series Sequence and series can be used to calculate the irrational number ap proximately For example7 1 The sequence aiil M 7 n 2 2 a positive real number 2 an71 an

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