M2 - RESULTS IN FIRST PART OF METHODS AND CALCULUS T.W.K ¨...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: RESULTS IN FIRST PART OF METHODS AND CALCULUS T.W.K ¨ ORNER Definition 1. Let a n , a ∈ R k . We say that a n → a as n → ∞ , if given ² > , we can find N ( ² ) such that || a n- a || < ² for all n > N ( ² ) . Theorem 2. (i) If a n → a and b n → b in R k then a n + b n → a + b as n → ∞ . (ii) If a n → a and b n → b in R (or C ) then a n b n → ab as n → ∞ . (iii) If a n → a as n → ∞ in R (or C ) and a 6 = 0 , a n 6 = 0 [ n = 1 , 2 ,... ] then a- 1 n → a- 1 as n → ∞ . There are many related definitions. Definition 3. (i) If f : R l → R k we say that f ( x ) → a as x → y if given ² > we can find a δ ( ² ) > such that, whenever < || x- y || < δ ( ² ) it follows that || f ( x )- a || < ² . (ii) Let a n ∈ R . We say that a n → ∞ if, given any K we can find N ( K ) such that a n > K for all n > N ( K ) . Axiom 4 (Fundamental Axiom of Analysis) . If a 1 , a 2 ,...is an increas- ing sequence in R and there exists an A ∈ R such that a n ≤ A for all n , then there exists an a ∈ R such that a n → a as n → ∞ . Definition 5. We say that the series a n in R k converges to the sum a if N X n =1 a n → a as N → ∞ . We write ∞ X n =1 a n = a . Lemma 6 (Absolute Convergence implies Convergence) . If a n ∈ R k and ∑ ∞ n =1 || a n || converges then ∑ ∞ n =1 a n converges. Lemma 7 (Comparison Test) . If a n ,b n ∈ R and ≤ a n ≤ b n then whenever ∑ ∞ n =1 b n converges ∑ ∞ n =1 a n must converge. 1 2 T.W.K ¨ ORNER Corollary 8 (Ratio Test) . If a n ∈ R , < a n and a n +1 /a n → l then (i) If l < 1 then ∑ ∞ n =1 a n converges. (ii) If l > 1 then ∑ ∞ n =1 a n diverges. Example 9. If a 2 n = 2- 2 n , a 2 n +1 = 2- 2 n- 2 then the ratio test fails but comparison with 2- n show that the series a n is convergent. Lemma 10 (Integral Test) . Suppose that f : [0 , ∞ ) → [0 , ∞ ) is a decreasing continuous function. Then if one of Z N f ( x ) dx and N X n =0 f ( n ) tends to a (finite) limit as N → ∞ so does the other. Corollary 11. ∑ ∞ n =1 n- p converges if and only if p > 1 . Note the failure of the ratio test for the series n- p . Example 12. If f ( x ) = 1- cos(2 πx ) then R N f ( x ) dx diverges and ∑ N n =0 f ( n ) converges as N → ∞ . Lemma 13 (Alternating Series Test) . (Not in syllabus.) Suppose that a n is a decreasing sequence of positive terms with a n → as n → ∞ . Then ∑ ∞ n =1 (- 1) n a n converges. Example 14. ∑ ∞ n =1 (- 1) n n- p converges for p > but only converges absolutely for p > 1 . Theorem 15 (Rearrangement of Positive Series) . Let σ : N → N be a bijection. If a n ≥ then, if the series a n converges, so does the rearranged series a σ ( n ) and ∞ X n =1 a n = ∞ X n =1 a σ ( n ) ....
View Full Document

This note was uploaded on 01/31/2011 for the course MATH 201b taught by Professor Groah during the Fall '08 term at UC Davis.

Page1 / 16

M2 - RESULTS IN FIRST PART OF METHODS AND CALCULUS T.W.K ¨...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online