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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 ) ....
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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.
 Fall '08
 Groah
 Math, Calculus

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