Question 81942
Since the decimal repeats, we can break up the series as


0.75+0.0075+0.000075...


So this represents a sequence. Notice the first term is 0.75, so {{{a[1]=0.75}}}. The common ratio (the constant factor) is then simply


{{{r=0.0075/0.75=1/100}}}


So altogether we have the sequence


{{{a[n]=0.75(1/100)^n}}} or  {{{a[n]=(3/4)(1/100)^n}}}


Notice if n=0 we get our first term 0.75, if n=1, we get 0.0075 etc.


So the series is simply:


{{{sum( (3/4)(1/100)^i, i=0, infinity )}}}