《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,-L< .A sequence that fails to converge to any finitenumber L is said to diverge, or to be divergent.ε-N Def:lim x, =LV>0,N>0, s.t. Vn>N=x, -L<n>0S 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 :