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11.1 Infinite SequencesA sequence is something like this:                   =                 ,   ,   ,···   +     =      ...
( )           ( )          =      =
(   ) ( + )           (   ) ( + )                  =              =
A sequence    {     }   has the limit        if     for every      >   there is a corresponding     integer       such tha...
What does it mean?                     =
Properties:For convergent sequence {   }    and   {   }        (   +    )=         +        (        )=        (   )=     ...
Some useful theorems: If                      for               and                         =             =      then     ...
Some useful examples:              =    if   >              =    if   =              =    if    < <               is diver...
Ex: Find           +
Ex: Find               +               =           +       +
Ex: Find               +               =           +           +               =                   +
Ex: Find               +               =           +           +               =                   +               =      ...
Ex: Find
Ex: Find                     /Since      =             =   (l’Hospital’s Rule)we have        = .
(   )Ex: Find
(       )Ex: Find           (       )Since                      =     =               (       )we have                    ...
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Calculus II - 20

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Stewart Calculus Section 11.1

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Calculus II - 20

  1. 1. 11.1 Infinite SequencesA sequence is something like this: = , , ,··· + = +
  2. 2. ( ) ( ) = =
  3. 3. ( ) ( + ) ( ) ( + ) = =
  4. 4. A sequence { } has the limit if for every > there is a corresponding integer such that for every > we have | |<We say { } is convergent, and write =or as
  5. 5. What does it mean? =
  6. 6. Properties:For convergent sequence { } and { } ( + )= + ( )= ( )= = ( )= · = if = ( )= if > , >
  7. 7. Some useful theorems: If for and = = then = . If | |= then = . If ( )= and ( )= then = . If = and ( ) is continuous at then ( ) = ( ).
  8. 8. Some useful examples: = if > = if = = if < < is divergent otherwise.
  9. 9. Ex: Find +
  10. 10. Ex: Find + = + +
  11. 11. Ex: Find + = + + = +
  12. 12. Ex: Find + = + + = + = = . +
  13. 13. Ex: Find
  14. 14. Ex: Find /Since = = (l’Hospital’s Rule)we have = .
  15. 15. ( )Ex: Find
  16. 16. ( )Ex: Find ( )Since = = ( )we have = .

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