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# 388 Review Sheet for MATH 401 at PSU

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Date Created: 02/06/15
Math 401 Introduction to Real Analysis Topics for Midterm I Review 1 Mathematical induction Given a sequence of statements P17 P2 P3 mathematical induction is a technique for prov ing that all of the statements are true Namely7 one has to show that i The rst statement P1 is true ii If Pk is true7 then also the following statement Pk is true 2 Upper bound supremum A set S C R is bounded above if there exists a number u such that u 2 a for all a E S In this case u is called an upper bound The smallest upper bound is called supremum and written sup S Theorem completeness of the real numbers If a set S is bounded above7 then it has a supremum To prove that u sup S7 one needs to show i u 2 a for every 9 6 S7 ii For every 6 gt 07 there exists a point a E S such that u 7 e lt 9 Notice that ii is certainly true if u E S 3 Sequences A sequence is a map from N into R It is usually denoted as 91 92 93 7 or xnn21 The sequence is bounded if all points M are contained in a bounded interval 11 It is monotone increasing if 91 g 92 g 933 g o A sequence can be de ned by directly assigning its values asn o Alternatively7 one can de ne the sequence by induction x the initial value 91 and then ii give a rule for computing zk1 from the previous value wk 4 Limits We say that the sequence 9 converges to a limit 6 and write lim 5 5 naoo if for every 6 gt 0 one can find a number N sufficiently large so that zn7 llt6 for all ngtN 1 This means that as n grows large the numbers 93 become closer and closer to 6 To prove that limWaco 93 6 one has to study the inequality 1 and show that it is satis ed for all integers n sufficiently large 5 Limit theorems Theorem sandwich lf 930721 340721 and 20721 are three sequences such that 93 g yn g 2 for every n 2 1 and if lim 93 Z lim 2 Hoe Hoe then we also have lim y z Hoe Theorem sums products quotients lf 9 and 3 are two sequences such that lim xna lim yny Hoe mace and c E B is a real number then lim znynzy7 lim czncz Hoe Hoe 11mzyzy lim ifyy O Hoe Hoe y y 6 Convergence criteria The following theorems guarantee that a sequence 9 converges even if we do not know precisely what the limit is o A sequence 930721 is bounded if there exists a number M large enough so that 93 6 PM M for all 71 Theorem monotone convergence Assume that the sequence is increasing so that 91 3 92 3 933 g Then the sequence converges to some limit a if and only if it is bounded In this case lim 93 supan n21 Hoe o A sequence 9 is a Cauchy sequence if7 for every 6 gt 0 one can nd an integer H large enough so that lzn 7 xml lt 6 for all mm gt lntuitively this means that7 when m7 n A 007 the numbers 93m 93 get closer and closer to each other Theorem Cauchy criterion A sequence 9 converges to some limit a if and only if it is a Cauchy sequence Theorem Bolzano Weierstrass If the sequence 9 is bounded7 then one can select integer numbers 711 lt 712 lt 713 lt such that the subsequence In xm 93m converges to some limit 7 Divergent sequences We say that the sequence 90721 tends to 00 and write limWaco 93 00 if for every arbitrarily large H E B there exists a number N such that 93 gt H for every integer n 2 N Theorem unbounded monotone sequences If the sequence 9 is monotone increasing and unbounded7 then lim asn 00 naoo Theorem comparison If 93 g yn for every n and if lim asn 00 then we also have naoo lim yn 00 Hoe 10 Series Given a sequence ma a we consider the in nite series 0 EM z1z2z3 n1 The corresponding sequence of partial sums is de ned as 81931 829319327 3k12 k7 If the sequence of partial sums 5k has a limit7 we say that the series is convergent We then de ne 0 E 93 lim 5k n1 kaoo ln general7 the series Z 93 converges if the terms 93 become smaller and smaller ie approach zero quickly enough The following theorems help in deciding if a series converges or not 3 0 Theorem necessary condition If the series E 93 converges then one must have lim 93 0 Hoe 721 Theorem comparison Assume 0 g 93 g yn for every 71 If the series 221 yn converges then the series 221 93 converges as well If the series 221 93 diverges then the series 221 yn diverges as well To use the above theorem it is useful to keep in mind that x the series 7 n 1 converges if p gt 1 diverges if p g 1 n1 0 the series E a 710 One should also remember the formula for the partial sums converges if lal lt 1 diverges if lal 2 1 7 k1 1aa2akl 1 7 a hence 17 k 1 Ea lim1aa2ak 11111 iflallt1 k700 k700 1 7 a 1 7 a n0 Theorem ratio test Assume that lim lzn l L lt1 woo 1m Then the series 221 93 converges Theorem alternating signs Let a1 2 a2 2 a3 2 be a decreasing sequence of positive 0 numbers such that lim an 0 Then the series with alternating signs 271 an is convergent n7ngtltgt n1 Basic Problems Using unique factorization prove that certain numbers such as are irrational Using mathematical induction7 prove that certain statements P17 P27 P3 are all true Decide whether a set S C R is bounded or not Find its supremum Decide if S has a maximum Prove that limWaco asn 6 using the de nition of limit7 or some theorem about limits Study a sequence 93 de ned inductively zk1 Check if it converges and nd its limit Check if a sequence is convergent or divergent7 using the de nition or a comparison method Decide if a series converges7 using the de nition or some convergence theorem Compute its sum7 in some special cases like a geometric series or a telescopic series

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