This document discusses power series representations of functions and their radii of convergence. It provides examples of finding the power series for functions like 1/(1+x) and sin(x), as well as the general process for determining the power series representation of a function and the radius of convergence based on the interval of convergence. It also discusses rules for differentiating and integrating term-by-term for power series whose functions have a given radius of convergence.

1. 11.9 Representations of
functions as power series
We already know that
= + + + + ···
=
converges to when < < .
On the contrary, can be expressed a
power series
= + + + + ···
=
when < < .

2. Ex: Find a power series representation for
+
and the radius of convergence.

3. Ex: Find a power series representation for
+
and the radius of convergence.
When | |< ,
= + + + ··· =
=

4. Ex: Find a power series representation for
+
and the radius of convergence.
When | |< ,
= + + + ··· =
=
Therefore,
= ( ) = ( )
+ = =

5. Ex: Find a power series representation for
+
and the radius of convergence.
When | |< ,
= + + + ··· =
=
Therefore,
= ( ) = ( )
+ = =
It converges when | |< , i.e. < < .

6. Ex: Find a power series representation for
+
and the radius of convergence.
When | |< ,
= + + + ··· =
=
Therefore,
= ( ) = ( )
+ = =
It converges when | |< , i.e. < < .
The radius of convergence is .

7. Ex: Find a power series representation for
and the radius of convergence.
+

8. Ex: Find a power series representation for
and the radius of convergence.
+
When | |< ,
= + + + ··· =
=

9. Ex: Find a power series representation for
and the radius of convergence.
+
When | |< ,
= + + + ··· =
=
Therefore,
( )
= = = +
+ + = =

10. Ex: Find a power series representation for
and the radius of convergence.
+
When | |< ,
= + + + ··· =
=
Therefore,
( )
= = = +
+ + = =
It converges when | / |< , i.e. < < .

11. Ex: Find a power series representation for
and the radius of convergence.
+
When | |< ,
= + + + ··· =
=
Therefore,
( )
= = = +
+ + = =
It converges when | / |< , i.e. < < .
The radius of convergence is .

12. Ex: Find a power series representation for
and the radius of convergence.
+

13. Ex: Find a power series representation for
and the radius of convergence.
+
( )
= = = +
+ + = =

14. Ex: Find a power series representation for
and the radius of convergence.
+
( )
= = = +
+ + = =
It converges when | / |< , i.e. < < .

15. Ex: Find a power series representation for
and the radius of convergence.
+
( )
= = = +
+ + = =
It converges when | / |< , i.e. < < .
The radius of convergence is .

16. Ex: Find a power series representation for
and the radius of convergence.
+
( )
= = = +
+ + = =
Therefore,
( ) ( ) +
= +
= +
+ = =
It converges when | / |< , i.e. < < .
The radius of convergence is .

17. Ex: Find a power series representation for ( )
and the radius of convergence.

18. Ex: Find a power series representation for ( )
and the radius of convergence.
( )=

19. Ex: Find a power series representation for ( )
and the radius of convergence.
( )=
= + + + + ···

20. Ex: Find a power series representation for ( )
and the radius of convergence.
( )=
= + + + + ···
= ··· +

21. Ex: Find a power series representation for ( )
and the radius of convergence.
( )=
= + + + + ···
= ··· +
= +
=

22. Ex: Find a power series representation for ( )
and the radius of convergence.
( )=
= + + + + ···
= ··· +
= +
=
Take = , we get = .

23. Ex: Find a power series representation for ( )
and the radius of convergence.
( )=
= + + + + ···
= ··· +
= +
=
Take = , we get = .
The radius of convergence is .

24. Theorem (term-by-term diff. and int.):
If the power series ( − ) has a radius
of convergence , then
∞
( )= = ( − )
is differentiable on the interval ( − , + )
and
∞ −
( )= = ( − )
+
∞ ( − )
( ) = + = +
The radii of both series are .

25. Ex: Find a power series representation for
( )
and the radius of convergence.

26. Ex: Find a power series representation for
( )
and the radius of convergence.
=
( )

27. Ex: Find a power series representation for
( )
and the radius of convergence.
=
( )
=( + + + + ···)

28. Ex: Find a power series representation for
( )
and the radius of convergence.
=
( )
=( + + + + ···)
= + + + ···

29. Ex: Find a power series representation for
( )
and the radius of convergence.
=
( )
=( + + + + ···)
= + + + ···
= ( + )
=

30. Ex: Find a power series representation for
( )
and the radius of convergence.
=
( )
=( + + + + ···)
= + + + ···
= ( + )
=
The radius of convergence is .