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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: = , { }= , , ,... = = = + = + = = + + = + + = ··· = = =