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Unformatted text preview: . [Hint: Add the relations p = 1, p 1 = 3, p k = ( k + 1) p k1 + p k2 for k = 2 , ..., n .] (b) Illustrate part (a) by calculating the numerator p 4 for the fraction [1; 2 , 3 , 4 , 5]. (7) If C k = p k /q k is the kth convergent of the simple continued fraction [ a ; a 1 , ..., a n ], show that q k ≥ 2 ( k1) / 2 , 2 ≤ k ≤ n. [Hint: Observe that q k ≥ a k q k1 + q k2 ≥ 2 q k2 .] 1 2 MATH 115B PROBLEM SET IV MAY 10, 2007 (8) If C k = p k /q k is the kth convergent of the simple continued fraction [ a ; a 1 , ..., a n ], and a > 0, show that p k p k1 = [ a k ; a k1 , ..., a 1 , a ] , and q k q k1 = [ a k ; a k1 , ..., a 2 , a 1 ] , Hint: In the ﬁrst case, notice that p k p k1 = a k + p k2 p k1 = a k + 1 p k1 p k2 ....
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 Fall '09
 Fractions, Continued fraction, simple continued fraction, simple continued fractions

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