Limit of A Sequence DEFINITION 11.3.1 LIMIT OF A SEQUENCE lim an =L n-→0o if for each e>0,there exists a positive integer K such that ifn≥K, then lan-LI<E. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence
Limit of a Sequence Example Show that 2vn imn+ =2 Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence Example Show that 2 lim 2 1 n n → n = +
Limit of A Sequence THEOREM 11.3.2 UNIQUENESS OF LIMIT If lim an =L and lim an =M, then L=M. n→00 1→0o Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence
Limit of A Sequence DEFINITION 11.3.3 A sequence that has a limit is said to be convergent.A sequence that has no limit is said to be divergent. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence
THEOREM 11.3.4 Every convergent sequence is bounded. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sons,Ine.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Every unbounded sequence is divergent. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved
Limit of A Sequence THEOREM 11.3.6 A nondecreasing sequence which is bounded above converges to the least upper bound of its range. A nonincreasing sequence which is bounded below converges to the greatest lower bound of its range. Sals,Hille,Etgen Calculus One and Several Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence
Limit of A Sequence Example Show that the sequence"+4)is convergent. Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence Example Show that the sequence ( ) is convergent. 1/ 3 4 n n n n a = +
Limit of A Sequence THEOREM 11.3.7 Let a be a real number.If an>L and b->M,then (①)an+bn→L+M, (ii)aan→aL, (iii)anbn-→LM. If,in addition,M≠0 and each b≠0，then 11 an L (w) and () M Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Limit of A Sequence
Example 7 n4-3n2+n+2 n3+7n Salas,Hille,Etgen Calculus:One and Several Varisbles Main Menu Copy right 2007 John Wiley Sors,Inc.All rights reserved
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved