   Chapter 11.2, Problem 78E

Chapter
Section
Textbook Problem

Graph the curves y = x n , 0 ≤ x ≤ 1 , for n = 0 , 1 , 2 , 3 , 4 , ... on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, shown in Example 8, that ∑ n = 1 ∞ 1 n ( n + 1 ) = 1

To determine

To graph:

The curves y=xn, 0 x 1, for n=0, 1, 2, 3, 4, .. on a common screen and give

a geometric demonstration of the fact that

n=11n(n+1)=1

by finding the areas between the successive curves.

Explanation

1) Concept:

Graph the given curves by using a calculator, and then find the area between the two curves using integration.

If sequence Sn is convergent and limnSn=s  exists as a real number, then the series n=1an is convergent and n=1an=s

2) Given:

y=xn, 0 x 1, for n=0, 1, 2, 3, 4, .. and

n=11n(n+1)=1

3) Calculation:

Graph of the given curves y=xn, 0x 1, for n=0, 1, 2, 3, 4, ..

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