   Chapter 11.P, Problem 6P

Chapter
Section
Textbook Problem

Find the sum of the series 1 + 1 2 + 1 3 + 1 4 + 1 6 + 1 8 + 1 9 + 1 12 + ... where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s.

To determine

To find:

The sum of the series

1+12+13+14+16+18+19+112+

Explanation

1) Concept:

Use the formula for the sum of geometric series to find the sum of the given series.

2) Given:

S=1+12+13+14+16+18+19+112+

3) Calculation:

Let the series

S=1+12+13+14+16+18+19+112+

Then every term in S is of the form 12m3n, m,n0,  and furthermore each term occurs only once.

So it can be written as

S=m=0n=012m3n=m=0n=012m13n

=m=012mn=013n

m=012m isa geometric series with initial term 1 and common difference 12

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