Class Notes (834,049)
Statistics (247)
STAT312 (22)
Lecture

# Sequences and Series.pdf

4 Pages
109 Views

School
Department
Statistics
Course
STAT312
Professor
Douglas Wiens
Semester
Fall

Description
106 21. Sequences and Series  Convergence of a sequence { } =1: We say that as if 0 ( | | ) (So the function ( ) = .)  Example: Let = , | | 1. Then 0. ( = log log | |.)  A monotonic, bounded sequence is convergent. (Proof: We did this earlier ...) Example: = 2, = 2 + . Since 1 +1 the function ( ) = 2 + is continuous, and = ( ), if the sequence is convergent to a +1 limit   then = ( ). Thus ( 2)( + 1) = 0. Claim: the sequence is increasing and bounded, hence it isconvergent and so = 2. For this, suppose that 0 2, as is true for = 1. 107 Then also +1 = 2 + 2 + 2 = 2  this shows that the sequence is bounded. Furthermore +1 2 + 2 2 + ( 2)( + 1) 0 which holds for [0 2]. By induction we con- clude that the sequence is increasing and bounded, hence convergent (t= 2). P  Series: Put = =1 , the partial sum of the series . We say that = if =1 =1 . P  Example: Geometric serie=0 for | | 1. We have X 1 +1 1 = = = =0 1 1  Numerous tests of convergence are available - see any text (e.g. your MATH 214 text) to review 108 some of them. We ha
More Less

Related notes for STAT312
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.