Notes03 - CSE 2320 Notes 3 Summations(Last updated 13:57 A3/P3 3.A G EOMETRIC S ERIES(review x k k =0 t å = x t 1 1 x 1 when x ¹ 1[Not hard to

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Unformatted text preview: CSE 2320 Notes 3: Summations (Last updated 3/26/12 13:57 A3/P3) 3.A. G EOMETRIC S ERIES (review) x k k =0 t å = x t +1- 1 x- 1 when x ¹ 1 [Not hard to verify by math induction] x k k =0 t å £ x k k =0 ¥ å = k ® ¥ lim x k- 1 x- 1 = 1 1- x when 0 < x <1 3.B. H ARMONIC S ERIES ln n £ H n = 1 k k =1 n å £ ln n + .577K £ ln n +1 As n approaches ∞ , H n H 2 n approaches? A. 1 B. 2 C. ln n D. n E. n ! F. ∞ 3.C. A PPROXIMATION BY I NTEGRALS For a monotonically increasing function x £ y Þ f x ( ) £ f y ( ) ( ): f ( x ) dx m- 1 n ò £ f ( k ) k = m n å £ f ( x ) dx m n +1 ò Since: f ( x ) dx k-1 k ò £ f ( k ) £ f ( x ) dx k k +1 ò in this situation. 2 3.D. B OUNDING S UMMATIONS U SING M ATH I NDUCTION AND I NEQUALITIES [Techniques are especially important for recurrences in notes 4] Show i 2 i =1 n å = Q n 3 ae è ç ö ø ÷ [Trivial to show using integration or i 2 i =1 n å = n n +1 ( ) 2 n +1 ( ) 6 .] a. Show O( n 3 ) (i) i 2 i =1 n =1 å = 1 £ cn 3 using any constant...
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This note was uploaded on 03/25/2012 for the course CSE 2320 taught by Professor Bobweems during the Spring '12 term at UT Arlington.

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Notes03 - CSE 2320 Notes 3 Summations(Last updated 13:57 A3/P3 3.A G EOMETRIC S ERIES(review x k k =0 t å = x t 1 1 x 1 when x ¹ 1[Not hard to

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