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### 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.

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/
Associated Topics:
High School Sequences, Series

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