Stewart Calculus Section 11.1

1. 11.1 Infinite Sequences
A sequence is something like this:
= , , ,···
+ = +

2. ( ) ( )
=
=

3. ( ) ( + ) ( ) ( + )
=
=

4. A sequence { } has the limit if
for every > there is a corresponding
integer such that
for every > we have | |<
We say { } is convergent, and write
=
or
as

5. What does it mean?
=

6. Properties:
For convergent sequence { } and { }
( + )= +
( )=
( )= =
( )= ·
= if =
( )= if > , >

7. Some useful theorems:
If for and
= =
then = .
If | |= then = .
If ( )= and ( )= then = .
If = and ( ) is continuous at
then ( ) = ( ).

8. Some useful examples:
= if >
= if =
= if < <
is divergent otherwise.

9. Ex: Find
+

10. Ex: Find
+
=
+ +

11. Ex: Find
+
=
+ +
=
+

12. Ex: Find
+
=
+ +
=
+
= = .
+

13. Ex: Find

14. Ex: Find
/
Since = = (l’Hospital’s Rule)
we have = .

15. ( )
Ex: Find

16. ( )
Ex: Find
( )
Since = =
( )
we have = .