Lec 25 - Sequences & Series

Recall:





 

Definition

Increasing, decreasing, monotonic, bounded sequence.

 

 Aside: Completeness Axiom

If a set of real numbers has an upper bound then there is a least upper bound.

 

 Theorem: Monotonic Sequence Theorem

Every bounded, monotonic sequence is convergent.

 

Proof:

1.  





By the axiom above, if it is bounded there is a least upper bound.



 

2.  





 

3.  





 

4.  



This implies



 

Example





Proof:



 

Play with the sequence

















…

 

1. Increasing: Proof by induction.
1. 



 

2. 

 

3. 











 

2. Bounded: Proof by induction.



1. 

 

2. 



 

3. 











 





 

3. To find the limit, take the limit of the formula.







 

Taking the limit,







 

Therefore,



 

 

 

 Series 11.2

 

Definition

An infinite series is an infinite sum of a sequence,



 

 

Definition





 

Definition







 





 

Definition

The geometric series is the sum of the geometric sequence.