Explanation Concept used: 1) The n t h harmonic number is the sum of the reciprocals of the first n natural numbers. H k = 1 1 + 1 2 + ... + 1 k 2) According to the “Principle of Mathematical Induction”, to prove that propositional function P ( n ) is true for all positive integers n , we have to perform the following two steps: BASIS STEP: We verify that P ( 0 ) is true. INDUCTIVE STEP: We show that the conditional statement P ( k ) → P ( k + 1 ) is true for all positive integers k > a . Proof: let P ( n ) , be the proposition that for some non-negative integer n , H 2 n ≤ 1 + n , is true. H 2 n = 1 1 + 1 2 + ... + 1 2 n ≤ 1 + n BASIS STEP: We verify that P ( 0 ) is true. Plug n = 0 , in above propositional function, we get 1 2 0 ≤ 1 + 0 1 ≤ 1 , which is true, thus P ( 0 ) is true. INDUCTIVE STEP: We start by assuming P ( n ) , is true for some positive integer k , i.e. we are assuming P ( k ) is true. This assumption is called “Inductive Hypothesis”. Now using this “Inductive Hypothesis”, we need to show that P ( k + 1 ) is also true. Assuming P ( k ) is true, we must have H 2 k = 1 1 + 1 2 + ... + 1 2 k ≤ 1 + k for some positive integer k ≥ 0 Next, for P ( k + 1 ) , we need to show H 2 k + 1 = 1 1 + 1 2 + ... + 1 2 k + 1 ≤ 1 + ( k + 1 ) , for some k such that ( k + 1 ) ≥ 0 Left hand side is, H 2 k + 1 = 1 1 + 1 2 + ... + 1 2 k + 1 = [ 1 1 + 1 2 + ... + 1 2 k ] + 1 2 k + 1 + 1 2 k + 2 + ... + 1 2 k + 1 ,, note that in 1 2 k + 1 + 1 2 k + 2 + ... + 1 2 k + 1 , there are 2 k terms H 2 k + 1 = 1 1 + 1 2 +

Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

8th Edition

ISBN: 9781259676512

Author: Kenneth H Rosen

Publisher: McGraw-Hill Education

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Chapter 5.1, Problem 29E

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For which non-negative integer n, H2n ≤ 1 + n is valid and prove the same.

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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991

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For which non-negative integer n , H 2 n ≤ 1 + n is valid and prove the same.

Solution Summary: The author explains how to prove that propositional function P(n) is true for all positive integers n.

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For which non-negative integer n, H2n ≤ 1 + n is valid and prove the same.

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For which non-negative integer n , H 2 n ≤ 1 + n is valid and prove the same.

Solution Summary: The author explains how to prove that propositional function P(n) is true for all positive integers n.

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To find: For which non-negative integer n, H2n ≤ 1 + n is valid and prove the same.

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Chapter 5 Solutions

To find:

For which non-negative integer n, H2n ≤ 1 + n is valid and prove the same.