   Chapter 11.4, Problem 40ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Use part (d) of Example 11.4.7 to show that n + n 2 + n 3 + ⋯ + n n is Θ ( n   ln   n ) .

To determine

To prove:

Use part (d) of Example 11.4.7 to show that n+n2+n3+...+nn is Θ(n ln n).

Explanation

Given information:

n+n2+n3+...+nn.

Proof:

ln n1+12+13+...+1n2 ln n whenever n3

Multiply each side of the inequality by n (assuming n3).

n ln nn(1+12+13+...+1n)2n ln n whenever n3

Use distributive property:

n ln nn+n2+n3+...+nn2n ln n whenever n3

As n3, each side of the inequality is nonnegative:

|n ln n||n+n2+n3+

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