   Chapter 11, Problem 23P

Chapter
Section
Textbook Problem

# Consider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit 0. Show that this series is convergent and the sum is less than 90.

To determine

To show: The series is convergent and the sum is less then 90.

Explanation

Result used:

(1) “The sum of the geometric series with initial term a and common ratio r is n=0arn=a1r .”

(2)The comparison test:

“Suppose that an and bn are series with positive terms

(a) If bn is convergent and anbn for all n ,then an is also convergent .

(b) If bn is divergent and anbn for all n ,then an is also divergent .”

Given:

The series whose terms are the reciprocals of the positive integer that can be written in base 10 notation without using the digit 0.

Calculation:

Consider S be the notation of the series,

Then group the terms whose terms are the reciprocals of the positive integer

S=(11+12+18+19)+(111++199)+(1111++1999)+=g1+g2+g3++gn+

Therefore, the group gn has 9 choices for each of the n digits in the denominator

That is gn has 9n terms and each term in gn less then 110n1

Therefore,

g</

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