Finding the Sum of an Infinite Series

Date: 03/05/2006 at 12:31:42
From: Alexandra
Subject: Finding the sum of series

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

Date: 03/05/2006 at 15:21:39
From: Doctor Vogler
Subject: Re: Finding the sum of series

Hi Alexandra,

Thanks for writing to Dr. Math.  That's a good question.  The trick is
to rearrange the terms.  First add up the ones with only 2's in the
denominator, as in

  1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

and use the formula for the sum of a geometric series.  Then add up
the ones with only one 3 in the denominator, as in

  1/3 + 1/6 + 1/12 + 1/24 + 1/48 + ...

using the same formula.  Then add up the ones with two 3's (i.e.
divisible by 9 but not 27) in the denominator, as in

  1/9 + 1/18 + 1/36 + 1/72 + ....

Keep going to get an infinite series of partial sums.

  (sum of 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...)
  + (sum of 1/3 + 1/6 + 1/12 + 1/24 + 1/48 + ...)
  + (sum of 1/9 + 1/18 + 1/36 + 1/72 + ...)
  + ...

Then try to add up the partial sums.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/