《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.4 The Limits of Sequences

Chapter1LimitsFunctionsandS 1.4 The Limits of Sequences

Chapter 1 Functions and Limits §1.4 The Limits of Sequences

I.Infinite Sequencea,az,"",an,isanorderedarrangementofrealnumbers.FormalDefinitionInfinitesequenceisafunctionwhosedomainisthesetof positive integers and whose range is a set of realnumbers.explicitformulaan =3n-2recursionformulaa, =1, a,=an-1+3, n≥2S1.4TheLimitsofSequences

§1.4 The Limits of Sequences I. Infinite Sequence a1 ,a2 ,  ,an ,  is an ordered arrangement of real numbers. Formal Definition Infinite sequence is a function whose domain is the set of positive integers and whose range is a set of real numbers. explicit formula an = 3n − 2 recursion formula a1 = 1, an = an−1 + 3, n  2

I. Infinite SeguenceFor instance:1n2a, =1+(-1)M3425an, =(-1)"5342n0.99,0.99,0.99, 0.99, ...a,=0.99Q: Do they converge to 1?S 1.4 The Limits ofSeguences

§1.4 The Limits of Sequences , 5 4 , 4 5 , 3 2 , 2 3 0, − − n a n n 1 = (−1) + , 5 4 , 4 5 , 3 2 , 2 3 0, n a n n 1 = 1+ (−1) , 5 4 , 4 3 , 3 2 , 2 1 0, n an 1 = 1− 0.99, 0.99, 0.99, 0.99,  an = 0.99 For instance: Q: Do they converge to 1? I. Infinite Sequence -1 0 1 • • ••• -1 0 1 • •• • • -1 0 1 • • • • • -1 0 1 •

Il. Limit of Infinite SequenceRelationship with the two limits1+n1+xf(x)nxV>0S 1.4TheLimitsofSequences

§1.4 The Limits of Sequences Relationship with the two limits x x f x + = 1 ( ) n n f n + = 1 ( )  −   +   1 1 0, n n II. Limit of Infinite Sequence

II. Limit of Infinite SequenceDef:The sequencef x.is said to converge to L, and we writelim x, = Ln->00if for each given number , there is a correspondingpositive number Nsuch that n>N=x,-L0,N>0, s.t. Vn>N=x, -L0S 1.4TheLimitsofSequences

§1.4 The Limits of Sequences II. Limit of Infinite Sequence Def: The sequence is said to converge to L, and we write if for each given number , there is a corresponding positive number N such that . A sequence that fails to converge to any finite number L is said to diverge, or to be divergent.  xn  xn L n = → lim  n  N  x − L   n   N  n  N  x − L   n =  0, 0,s.t. → xn L n lim − N Def :

Il. Limit of Infinite Sequencelim x, =a≤V>0,3N>0, s.t. Vn>N=x, -al N, all the points x, are in the neighborhoodU(a,e) where there are finite points (at most N points)not in .S 1.4TheLimitsofSeguences

§1.4 The Limits of Sequences II. Limit of Infinite Sequence Geometric interpretation x 1 x 2 x xN +1 xN +2 x3 2 a −  a +  a x4   N  n  N  x − a   s.t. n =  0, 0, → xn a n lim When , all the points xn are in the neighborhood where there are finite points (at most N points) not in . U(a, ) n  N

II. Limit of Infinite SequenceExample11. Prove that the limit of the sequence 1,q,q',..",q"-l,is 0, provided that q 1+logg 8.We choose N =[1+ log/a ] ,Then n> N implies thatThat is lim qn-l = 0.S1.'4TheLimitsofSequences

§1.4 The Limits of Sequences Example 1 0 1. 1. 1, , , , , 2 1  − q q q q n is ,provided that Prove that the limit of the sequence   Proof Let be given.  that is 1 log  . q n  + If we want , − =   − − 1 1 0 n n q q Then n  N implies that   n−1 q We choose    , N q = 1+ log That is lim 0 . 1 = − → n n q II. Limit of Infinite Sequence

I. Limit of Infinite SequenceExample 12. Prove thatlim(Vn+1-Vn)= 0n>oS 1.4 TheLimits ofSeguences

§1.4 The Limits of Sequences lim( +1 − ) = 0 → n n n 2.Prove that Example 1 II. Limit of Infinite Sequence

Ill. Properties of the Limit1.UniquenessTh: Let x, →a,x, →b(n →oo), then a = b.2.BoundednessTh: Let x, →a(n →oo), then x, is boundedS 1.4TheLimitsofSeguences

§1.4 The Limits of Sequences 1. Uniqueness x a, x b(n ), a b. Th: Let n → n → →  then = 2. Boundedness ( ), then is bounded. xn → a n →  xn Th: Let III. Properties of the Limit

Ill. Properties of the Limit1.UniquenessTh: Let lim f(x) = A,lim f(x) = B,then A = B.x→a-2. Local boundedness1Th:Let lim f(x) = A, then 3S > 0, when x e U(a,S),x→a3M > 0, f(x)|≤ M.S 1.4TheLimitsofSeguences

§1.4 The Limits of Sequences 1. Uniqueness lim f (x) A,lim f (x) B, A B. x a x a = = = → → Th: Let then 2. Local boundedness Th: 0, ( ) . lim ( ) , 0, ( , ), 0 M f x M f x A x U a x a    =    → Let then  when  III. Properties of the Limit