   Chapter 7.8, Problem 48E

Chapter
Section
Textbook Problem

# (a) If g ( x ) = 1 / ( x − 1 ) , use your calculator or computer to make a table of approximate values of ∫ 2 t g ( x )   d x for t = 5 , 10 , 100 , 1000 ,  and 10, 000 . Does it appear that ∫ 2 ∞ g ( x )   d x is convergent or divergent?(b) Use the Comparison Theorem with f ( x ) = 1 / x to show that ∫ 2 ∞ g ( x )   d x is divergent.(c) Illustrate part (b) by graphing f and g on the same screen for 2 ≤ x ≤ 20 . Use your graph to explain intuitively why ∫ 2 ∞ g ( x )   d x is divergent.

To determine

( a)

To make:

A table of approximate values of 2tg(x)dx using calculator or computer and state if it appear convergent or divergent.

Explanation

Given:

g(x)=1x1

Consider g(x)=1x1

Now, put t=5,10,100,1000,10000 and get the table

To determine

(b)

To show:

That the integral 2g(x)dx is divergent using comparison theorem.

To determine

(c)

To illustrate:

Part (b) by graphing  f and g on the same screen for 2x20 and explain why 2g(x)dx is convergent.

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