11.1 Infinite SequencesSome useful examples:            =    if     >             =   if     =             =   if     < < ...
Properties:For convergent sequence {   }    and   {   }        (   +    )=         +        (        )=        (   )=     ...
Some useful theorems: If                      for               and                         =             =      then     ...
A sequence { } is called increasing if   < + for all           , it is calleddecreasing if   > + for all             . It ...
Ex: Show that   =       is decreasing.                    +
Ex: Show that       =          is decreasing.                        +We can show that                        +           ...
Ex: Show that       =          is decreasing.                         +We can show that                        +          ...
A sequence   {    }  is bounded above if there isa number         such that                      ,It is bounded below if t...
If for every number        there is an integersuch that              >     when      >we say                       =     ....
11.2 Infinite SeriesAn infinite series regarding a sequence {   }is something like               +       +     + ···which ...
∞Given a series        == +                   +   + ···let  denotes its partial sum:          =       =        +       + ·...
Ex:   =   ,   {   }=   ,   , ,...
Ex:   =     ,   {   }=   ,   , ,...      =     =      =     +   =    +   =      =     +   +    =   +    +   =      ···
Ex:   =     ,   {   }=   ,   , ,...      =     =      =     +   =    +   =      =     +   +    =   +    +   =      ···    ...
Ex:       =         ,   {   }=   ,   , ,...          =     =          =     +       =    +   =          =     +       +   ...
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Calculus II - 21

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Stewart Calculus Section 11.1

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  • Calculus II - 21

    1. 1. 11.1 Infinite SequencesSome useful examples: = if > = if = = if < < is divergent otherwise.
    2. 2. Properties:For convergent sequence { } and { } ( + )= + ( )= ( )= = ( )= · = if = ( )= if > , >
    3. 3. Some useful theorems: If for and = = then = . If | |= then = . If ( )= and ( )= then = . If = and ( ) is continuous at then ( ) = ( ).
    4. 4. A sequence { } is called increasing if < + for all , it is calleddecreasing if > + for all . It iscalled monotonic if it is either increasing ordecreasing.
    5. 5. Ex: Show that = is decreasing. +
    6. 6. Ex: Show that = is decreasing. +We can show that + > + ( + ) +
    7. 7. Ex: Show that = is decreasing. +We can show that + > + ( + ) +Or we can show that ( )= is decreasing. +
    8. 8. A sequence { } is bounded above if there isa number such that ,It is bounded below if there is a numbersuch that ,If it is bounded above and below it is abounded sequence.Monotonic sequence theorem:Every bounded monotonic sequence isconvergent.
    9. 9. If for every number there is an integersuch that > when >we say = .If for every number there is an integersuch that < when >we say = .It is still divergent, but in a special way.
    10. 10. 11.2 Infinite SeriesAn infinite series regarding a sequence { }is something like + + + ···which is often denoted as =Not all series makes sense! For example, + + + + ···is not meaningful.
    11. 11. ∞Given a series == + + + ···let denotes its partial sum: = = + + ··· + =then { } is a new sequence. If it isconvergent: = ∞we call the series = convergent andwrite = . =
    12. 12. Ex: = , { }= , , ,...
    13. 13. Ex: = , { }= , , ,... = = = + = + = = + + = + + = ···
    14. 14. Ex: = , { }= , , ,... = = = + = + = = + + = + + = ··· =
    15. 15. Ex: = , { }= , , ,... = = = + = + = = + + = + + = ··· = = =

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