   Chapter 11.P, 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 formed by the given information is convergent, and the sum is less than 90

Explanation

1) Concept:

i. The Comparison Test

Suppose an and bn are series with positive terms.

If n=1bn is convergent and anbn for all n, then an is also convergent.

If n=1bn is divergent and anbn for all n, then an is also divergent.

ii. The sequence rn is convergent if -1<x1 and divergent for all other values of r

limnrn=0  if -1<r<11            if  r=1

2) Calculation:

It is given that the terms of the series are the reciprocals of the positive integers that can be written in base 10 notation without using the digit 0.

That is, the series will be

S=11+12+13++18+19+111+112++198+199+1111++1999+

Assume that

g1=11+12+13++18+19

g2=111+112++198+199

g3=1111++1999

And so on

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