A power series is a series of the form Σan(x-c)n where an are the coefficients of the series. The sum of the series is a function of x. A power series centered at c or about c is of the form Σan(x-c)n. For a given power series, there are three possibilities for convergence: it converges only at c, converges for all x, or converges when |x-c|<R and diverges when |x-c|>R, where R is the radius of convergence.

1. 11.8 Power Series
A power series is a series of the form
= + + + + ···
=
{ } are called the coefficient of the series.
For each fixed , the series can be
convergent or not.
The sum of the series is a function of .

2. Ex: For what values of is the series
convergent? =

3. Ex: For what values of is the series
convergent? =
It is a geometric series. When < < ,
the series converges.

4. Ex: For what values of is the series !
convergent? =

5. Ex: For what values of is the series !
convergent? =
+
+ ( + )!
= = ( + )| |
!

6. Ex: For what values of is the series !
convergent? =
+
+ ( + )!
= = ( + )| |
!
+
= ( + )| | = except =

7. Ex: For what values of is the series !
convergent? =
+
+ ( + )!
= = ( + )| |
!
+
= ( + )| | = except =
The series converges when = .

8. Ex: For what values of is the series
convergent? =

9. Ex: For what values of is the series
convergent? =
| |
| |=

10. Ex: For what values of is the series
convergent? =
| |
| |=
| |= for any

11. Ex: For what values of is the series
convergent? =
| |
| |=
| |= for any
The series converges for any .

12. ( )
Ex: For what values of is the series
convergent? =

13. ( )
Ex: For what values of is the series
convergent? =
+
+ ( ) ( )
= = | |
+ +

14. ( )
Ex: For what values of is the series
convergent? =
+
+ ( ) ( )
= = | |
+ +
+
=| |

15. ( )
Ex: For what values of is the series
convergent? =
+
+ ( ) ( )
= = | |
+ +
+
=| |
The series converges when | |< i.e. < <

16. ( )
Ex: For what values of is the series
convergent? =
+
+ ( ) ( )
= = | |
+ +
+
=| |
The series converges when | |< i.e. < <
In fact, it is convergent when < .

17. ∞
A series of the form = ( − ) is called a
power series centered at or a power series about .

18. ∞
A series of the form = ( − ) is called a
power 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. Ex: Find the radius of convergence of the series
( + )
+
=

20. Ex: Find the radius of convergence of the series
( + )
+
=
+
+ ( + )( + ) ( + ) + | + |
= + +
=

21. Ex: Find the radius of convergence of the series
( + )
+
=
+
+ ( + )( + ) ( + ) + | + |
= + +
=
+ | + |
=

22. Ex: Find the radius of convergence of the series
( + )
+
=
+
+ ( + )( + ) ( + ) + | + |
= + +
=
+ | + |
=
The series converges when | + |< i.e. < <

23. Ex: Find the radius of convergence of the series
( + )
+
=
+
+ ( + )( + ) ( + ) + | + |
= + +
=
+ | + |
=
The series converges when | + |< i.e. < <
The radius of convergence is .