Stewart Calculus Section 11.5&6

1. 11.5 Alternating Series
11.6 The Ratio and Root Tests
An alternating series is a series whose terms
are alternately positive and negative. For
example:
( )
= + + ···
=

2. The Alternating Series Test:
If the alternating series
( ) = + ··· >
=
satisfies + and =
then the series is convergent.

3. ( )
Ex: Test series for convergence.
=
+

4. ( )
Ex: Test series for convergence.
=
+
= >
+

5. ( )
Ex: Test series for convergence.
=
+
= >
+
=

6. ( )
Ex: Test series for convergence.
=
+
= >
+
=
decreasing?

7. ( )
Ex: Test series for convergence.
=
+
= >
+
=
decreasing?
When , it is decreasing (check by derivative).

8. ( )
Ex: Test series for convergence.
=
+
= >
+
=
decreasing?
When , it is decreasing (check by derivative).
( )
So is convergent.
=
+

9. Notice that
= + + + + ···
=
is divergent but
( )
= + + ···
=
is convergent.
∞
A series = is called absolutely
∞
convergent if = | | is convergent.
It is called conditionally convergent if it is
∞
convergent but = | | is divergent.

10. Theorem: Absolutely convergent series is
convergent.
| | | |
< is convergent because <
=
| | | |

11. Theorem: Absolutely convergent series is
convergent.
Ex: Determine if is convergent.
=
| | | |
< is convergent because <
=
| | | |

12. Theorem: Absolutely convergent series is
convergent.
Ex: Determine if is convergent.
=
It has both positive and negative terms,
but not alternating.
| | | |
< is convergent because <
=
| | | |

13. Theorem: Absolutely convergent series is
convergent.
Ex: Determine if is convergent.
=
It has both positive and negative terms,
but not alternating.
| | | |
< is convergent because <
=
| | | |
so is absolutely convergent,
=

14. Theorem: Absolutely convergent series is
convergent.
Ex: Determine if is convergent.
=
It has both positive and negative terms,
but not alternating.
| | | |
< is convergent because <
=
| | | |
so is absolutely convergent,
=
so is convergent.
=

15. The Ratio Test:
+
If = < , then the series
=
is absolutely convergent (and thus convergent).
+
If = > or = , then the series
is divergent.
=
+
If = , no conclusion can be drawn.

16. The Root Test:
If | |= < , then the series
=
is absolutely convergent (and thus convergent).
If | |= > or = , then the series
is divergent.
=
If | |= , no conclusion can be drawn.

17. Ex: Test the series for convergence.
=

18. Ex: Test the series for convergence.
=
+ ( + ) +
= +
=
So is (absolutely) convergent.
=

19. +
Ex: Test the series for convergence.
=
+

20. +
Ex: Test the series for convergence.
=
+
+
| |=
+
+
So is (absolutely) convergent.
=
+