   Chapter 5, Problem 72RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Evaluate lim n → ∞ 1 n [ ( 1 n ) 9 + ( 2 n ) 9 + ( 3 n ) 9 + ⋯ + ( n n ) 9 ]

To determine

To calculate: The value of the function limn1n[(1n)9+(2n)9+(3n)9+....+(nn)9].

Explanation

Given information:

The function is limn1n[(1n)9+(2n)9+(3n)9+...+(nn)9] (1)

Apply theorem 4 of Definite integral:

abf(x)dx=limni=1nf(xi)Δx

Consider Δx=ban and xi=a+iΔx.

Here, the upper limit is b, the lower limit is a, width of subintervals is Δx, and number of intervals is n.

Consider that theorem 4 is valid under following condition:

• The function f is integrable on (a,b).

Rearrange Equation (1) as shown below.

limn1n[(1n)9+(2n)9+(3n)9+..

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