##### School: University of California Riverside

##### Department: Applied Mathematics

##### Course: First Year Calculus

##### Professor: Steven Gindi

##### Term: Spring 2019

##### Tags: Math, math009C, Calc, Calculus, sequence, Series, Convergence, bounded, Unbounded, and monotonic

##### Name: Math 009C Week 1

##### Description: Discusses sequences and introduces series

##### Uploaded: 04/09/2019

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Unformatted text preview: SEQUENCES SERIES Sequences Factorial .4! refers to the # 4.3.2.1 : 24 n!: n(n-1).(n-2)... 21 O!! Def: Sequence Informal: a list of #s Formal Sequence is a function acn) whose domain is N. The range of a sequence is the set of all distinct values of acn) 4 Terms of a sequence are the values a(9), a[2)... denoted asa, az .. 4 Sequence ain) is denoted and Example: an. 3n alo) a( 04 a (2) 7 Def : Limit of Sequence, Convergent, Divergent Let tans be a sequence let L be a real *. Given any E > O, if an m can be found such that lan-L1 & for all nim, then we say the limit of an as n approaches o is L, denoted nto an If no an exists, we say the sequence converges, otherwise the sequence diverges As n the sequence stays arbitrarily close to L Example: consider can waaz'ayasa... the rest is zero. Does this converge to zero? Thm: Limit of a Sequence Let {2,} be a sequence let f(x) be a function whose domain contains positive real #s where finan for all nin N. If **fcx) L, then mom an'l Example: let {an be defined as an in . Does it converge? - I COSx) ! for all x by Squeeze Thm Jim COSC) thus {n} converges to zerolimfa 2009 (x) pal's Rule Framework If *****g(x) + fyr) or 8 Then f(x) g(x) are diferentiable tim i (x) 20 'cx) Example lim 3x:7x + 5 lim 6x07 - Y+08 4x11x15 Example: let {a} be defined by an Inin). Does it converge! Lim In ) lim Converges to zero Def : Bounded Unbounded Sequences A sequence land is bounded if there exist real #s m. such that m an M for all n in N A sequence is Unbounded if it's not bounded A sequence and is bounded above if there exists an M such that a M for all n in N is bounded below if there exists an m such that mean for all n in IN Example: an sinn) or an cos(n) -1 sin in) 1 1 COSCn) ! both are bounded Example annt is bounded below .because on? o, but is not bounded above since it gets arbitrarily large Example: let an em for n o land is bounded below because an? check for upper bound a limbilot o On converges to zero! Note: Let {an be a convergent sequenceta is bounded Def: Monotonic Sequences A sequence a, is monotonically increasing if an anel for all a, az az !...an sanoi.. 2. A sequence tans is monotonically decreasing if an? Ons for all n a, a, a, ... an ? anti... 3. A sequence is monotonic if it's monotonically increasing or decreasing Example: an' is monotonically cecreasingExample: an an anti antiano anal: ht : 2n2 n) n+ 2 anu-an n.+2n+2 -1 n2n. (4 (622n+2) - (nt (n+2) ent2)(n+1) : min. 2n. 2) (n2n2) CR. 2nnt2) (n+ 2)(n+1) n.2n2n.n.2n. 2-n'- 2n'.n.2. (n+2)(n+1) sn +3n n+2)(n+1) all terms are positive so: anne - ano monotonically increasing Example: let an' (t) for n? ay24 not monotonic but converges @ zero JL Infinite Series Def: Infinite Series, nth Partial Sums, Convergence, Divergence Let. and be a sequence 1. The sum n an is a series/infinite series 2. Let 5 2, Q. The sequence {n} is the sequence of nth partial sums of Cans 3. If the sequence is converges to L, we say the series in an Converges to L, Written anul anil 4. If the sequence {n} diverges, the series Enl an diverges Example: Init 72 5,a:12 42. S, Q.2, a, Seems like Sn11