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Lecture

Sequences and Series.pdf

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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
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