Find \(1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{36} + \dotsb,\) where we alternately multiply by 1/2 and 1/3 to get the next term.

Guest Feb 18, 2019

#1**+2 **

Are you saying that you multiply this finally number in the set of digits by 1/2 or 1/3 ? or multiply the previous fraction by those to fractions.

HiylinLink Feb 18, 2019

#2**+3 **

We have

1 + 1/2 + 1/6 + 1/12 + 1/36 + 1/72 + ......

Notice that we can split this up like so

1 + 1/6 + 1/36 + ......+

1/2 + 1/12 + 1/72 + ......+

The sum of the first series is 1 / (1 - 1/6) = 1 / (5/6) = 6/5

The sum of the second series is (1/2) / ( 1 - 1/6) = (1/2) /(5./6) = 6/10 = 3/5

So....the sum of the series is 6/5 + 3/5 = 9/5

CPhill Feb 18, 2019