A sequence may be specified by an explicit formula or a recurrence relation: although sequences are not always defined these ways. An = (cid:4666) (cid:2869)(cid:4667)(cid:2870: {(cid:2869)(cid:2869)(cid:2868), (cid:2869)(cid:2869)(cid:2868)(cid:2868), (cid:2869)(cid:2869)(cid:2868)(cid:2868)(cid:2868) , { (cid:2869)(cid:2870),(cid:2869)(cid:2872), (cid:2869)8,(cid:2869)(cid:2869)6 } A recurrence relation for a sequence is given by. Defining an+1 = f(an) for n 1. If the terms of sequence an approach a unique number l as n increases, lim (cid:1853) converges to l. if there is no such l, an diverges. Series sum of the numbers in a sequence. Let {an}, {bn}, {cn} be sequences such that an bn cn for all n larger than some m. Theorem suppose f is a function such that f(n) = an for all integers n. if lim (cid:4666)(cid:4667)= then lim (cid:1853)= The following sequences are ordered according to growth rates as n : {ln(cid:2870)},{(cid:3043)},{(cid:3043)ln(cid:3045)},{(cid:3043)+(cid:3044)},{(cid:1854)},{! Every bounded, monotone sequence converges: monotone: strictly increasing or decreasing.
