   Chapter 11.4, Problem 37E

Chapter
Section
Textbook Problem

The meaning of the decimal representation of a number 0. d 1 d 2 d 3 ... (where the digit d i is one of the numbers 0, 1, 2,…, 9) is that 0. d 1 d 2 d 3 d 4 ... = d 1 10 + d 2 10 2 + d 3 10 3 + d 4 10 4 + ... Show that this series always converges.

To determine

To show:

The series 0.d1d2d3d4=d110+d2102+d3103+d4104+   always converges.

Explanation

1) Concept:

an and bn are both convergent series with positive terms, then if

i) bn is convergent and anbn  then an  is also convergent.

2) Given:

0.d1d2d3d4=d110+d2102+d3103+d4104 where each di is one of the numbers 0,1,29

3) Calculation:

The given series is

0.d1d2d3d4=d110+d2102+d3103+d4104

Each di is one of the numbers 0,1,29

Therefore,

dn10n910

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