tailieunhanh - A Collection of Limits

To help you have more documents to serve the needs of learning and research, invite you to consult the content "A Collection of Limits" below. Contents World raw materials to you: Short theoretical introduction, problems, solutions. Hope this is useful references for you. | VMF A Collection of Limits VMF Contents 1 Short theoretical introduction 1 2 Problems 12 3 Solutions 23 2 VMF Chapter 1 Short theoretical introduction Consider a sequence of real numbers an n 1 and l 2 R. We ll say that l represents the limit of an n 1 if any neighborhood of l contains all the terms of the sequence starting from a certain index. We write this fact as lim an l n i or an l. We can rewrite the above definition into the following equivalence lim an l 8 V 2 V l 9 nV 2 N such that 8 n nV an 2 V. n i One can easily observe from this definition that if a sequence is constant then it s limit is equal with the constant term. We ll say that a sequence of real numbers an n 1 is convergent if it has limit and lim an 2 R or divergent if it doesn t have a limit or if it has the limit nn equal to 1. Theorem If a sequence has limit then this limit is unique. Proof Consider a sequence an n 1 c R which has two different limits l0 l 2 R. It follows that there exist two neighborhoods V0 2 V l0 and V00 2 V l00 such that V0 n V00 . As an l0 9 n0 2 N such that 8 n n an 2 V0. Also since an l00 9 n00 2 N such that 8 n n00 an 2 V00. Hence 8 n maxfn0 n00g we have an 2 V0 V00 . Theorem Consider a sequence of real numbers an n 1. Then we have i lim an l 2 R 8 0 3 ne 2 N such that 8 n ne an l .