Calculus II - 21
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This document discusses infinite sequences and infinite series. Some key points: 1. It provides examples of determining if an infinite sequence converges or diverges based on the limit of its terms. 2. Properties of operations on convergent sequences are given, such as sums, products, and compositions. 3. Theorems regarding convergence of sequences are stated, such as if the limit of the sequence of partial sums exists. 4. Examples are given of showing monotonicity (increasing or decreasing behavior) of specific sequences. 5. Boundedness of sequences is defined, and the monotonic sequence theorem is stated. 6. Infinite series are introduced as the sum of the terms of a
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- 1. 11.1 Infinite Sequences Some useful examples: = if > = if = = if < < is divergent otherwise.
- 2. Properties: For convergent sequence { } and { } ( + )= + ( )= ( )= = ( )= · = if = ( )= if > , >
- 3. Some useful theorems: If for and = = then = . If | |= then = . If ( )= and ( )= then = . If = and ( ) is continuous at then ( ) = ( ).
- 4. A sequence { } is called increasing if < + for all , it is called decreasing if > + for all . It is called monotonic if it is either increasing or decreasing.
- 5. Ex: Show that = is decreasing. +
- 6. Ex: Show that = is decreasing. + We can show that + > + ( + ) +
- 7. Ex: Show that = is decreasing. + We can show that + > + ( + ) + Or we can show that ( )= is decreasing. +
- 8. A sequence { } is bounded above if there is a number such that , It is bounded below if there is a number such that , If it is bounded above and below it is a bounded sequence. Monotonic sequence theorem: Every bounded monotonic sequence is convergent.
- 9. If for every number there is an integer such that > when > we say = . If for every number there is an integer such that < when > we say = . It is still divergent, but in a special way.
- 10. 11.2 Infinite Series An infinite series regarding a sequence { } is something like + + + ··· which is often denoted as = Not all series makes sense! For example, + + + + ··· is not meaningful.
- 11. ∞ Given a series == + + + ··· let denotes its partial sum: = = + + ··· + = then { } is a new sequence. If it is convergent: = ∞ we call the series = convergent and write = . =
- 12. Ex: = , { }= , , ,...
- 13. Ex: = , { }= , , ,... = = = + = + = = + + = + + = ···
- 14. Ex: = , { }= , , ,... = = = + = + = = + + = + + = ··· =
- 15. Ex: = , { }= , , ,... = = = + = + = = + + = + + = ··· = = =
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