Question 1098493
{{{1+1/3+4/9+8/9+"..."}}} is not exactly a geometric series
{{{1+2/3+4/9+8/9+16/27+32/81+"..."}}} is a geometric series
with first term {{{b=1}}} and common ratio {{{r=2/3}}} .
For any geometric series with {{{r<1}}} ,
the sum to infinity can be calculated as
{{{S=b/(1-r)}}} ,
so {{{1+2/3+4/9+8/9+"..."=1/(1-2/3)=1/(1/3)=3}}}
 
If what you really want to calculate is
{{{1+1/3+4/9+8/9+16/27+32/81+"..."=(-1/3)+(1+2/3+4/9+8/9+16/27+32/81+"...")}}} ,
then
{{{1+1/3+4/9+8/9+16/27+32/81+"..."=(-1/3)+3=2&2/3=8/3}}} .