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# Calculus II - 26

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Stewart Calculus 11.8

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• ### Calculus II - 26

1. 1. 11.8 Power SeriesA power series is a series of the form = + + + + ··· ={ } are called the coefficient of the series.For each fixed , the series can beconvergent or not.The sum of the series is a function of .
2. 2. Ex: For what values of is the series convergent? =
3. 3. Ex: For what values of is the series convergent? =It is a geometric series. When < < , the series converges.
4. 4. Ex: For what values of is the series ! convergent? =
5. 5. Ex: For what values of is the series ! convergent? = + + ( + )! = = ( + )| | !
6. 6. Ex: For what values of is the series ! convergent? = + + ( + )! = = ( + )| | ! + = ( + )| | = except =
7. 7. Ex: For what values of is the series ! convergent? = + + ( + )! = = ( + )| | ! + = ( + )| | = except =The series converges when = .
8. 8. Ex: For what values of is the series convergent? =
9. 9. Ex: For what values of is the series convergent? = | | | |=
10. 10. Ex: For what values of is the series convergent? = | | | |= | |= for any
11. 11. Ex: For what values of is the series convergent? = | | | |= | |= for anyThe series converges for any .
12. 12. ( )Ex: For what values of is the series convergent? =
13. 13. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + +
14. 14. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + + + =| |
15. 15. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + + + =| |The series converges when | |< i.e. < <
16. 16. ( )Ex: For what values of is the series convergent? = + + ( ) ( ) = = | | + + + =| |The series converges when | |< i.e. < <In fact, it is convergent when < .
17. 17. ∞A series of the form = ( − ) is called apower series centered at or a power series about .
18. 18. ∞A series of the form = ( − ) is called apower series centered at or a power series about . ∞Theorem: For a given power series = ( − )there are only three possibilities: The series converges only when = . The series converges for all . The series converges when | |< and diverges when | |> is called the radius of convergence. Anything can happen = ± .
19. 19. Ex: Find the radius of convergence of the series ( + ) + =
20. 20. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + =
21. 21. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + = + | + | =
22. 22. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + = + | + | = The series converges when | + |< i.e. < <
23. 23. Ex: Find the radius of convergence of the series ( + ) + = + + ( + )( + ) ( + ) + | + | = + + = + | + | = The series converges when | + |< i.e. < < The radius of convergence is .