# calculus

Jun 18th, 2016
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notes on sequences and series

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SEQUENCE AND SERIESSEQUENCESA sequence can be thought of as a list of numbers written in a definite order a1 , a2 , a3 ,, an ,. Thethnumber a1 is called the first term, a2 is the second term and in general an is the n term. Noticethat for every positive integer n there is a corresponding number an and so a sequence can bedefined as a function whose domain is the set of positive integers.Notation: The sequence a1 , a2 , a3 , is also denoted by an or an n 1, an ,EXAMPLESth1. Some sequences can be defined by giving a formula for the n terma) n n 1 n 1SOLUTIONan n1 2 3 4 , , , ,n 1 2 3 4 5,n, n 1 (1) n ( n 1) b) 3nn 1SOLUTIONan c)(1) n (n 1) 1 2 3 4 , , , ,3n2 3 4 5n3,n, n 1 n 3SOLUTIONan n 3, n 3 0,1, 2, 3,, n 3, n d) cos6 n0SOLUTIONan cos2.n,n 06 3 1, , 0,1, 2 2, cosn, 635Find a formula for the general term an of the sequence , assuming that the pattern of the first few terms continuesSOLUTION34567a1 , a2 , a3 , a4 , a5 525125625312514 567,,,, 25 125 625 3125 Notice that the numerators of these fractions start with 3 and

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