Find the sum of the series 1+1/2+1/10+1/20+1/100..., where we alternately multiply by 1/2 and 1/5 to get successive terms.

1 + 1/2 + 1/10 + 1/20 + 1/100 + 1/200 +......

Notice that we can actually write this as the sum of two infinite series...........

 [1 + 1/10 + 1/100 +....] + [ 1/2 + 1/20 + 1/200 +....]

The first series has 1 as its first term and a common ratio of 1/10

The second series has 1/2 as its first term and also has a common ratio of 1/10

So  the sum  of these two series can be expressed as

1 / [ 1 - 1/10]    +   (1/2)/ [1 - 1/10]  =  

[ ! + 1/2] / [1 - 1/10 ] = 

[ 3/2] / [ 9/10] =

[3/2] * [10/9] =  

30 / 18   = 

5 / 3