Question 1074919
{{{S[1]=1+x^3+x^6+x^9+"..."=sum( x^(3k),k=0, infinity )=1/(1-x^3)}}}
is one of the infinite geometric series hinted at.
The first term is {{{1}}} and the common ratio is {{{x^3}}} .
The other two series are
{{{S[2]=-x-x^4-x^7-x^10+"..."=sum( -x^(3k+1),k=0, infinity )=-x/(1-x^3)}}} ,
with first term {{{-x}}} , and
{{{S[3]=-x^2-x^5-x^8-x^11-"..."=sum( -x^(3k+2),k=0, infinity )=-x^2/(1-x^3)}}} ,
with first term {{{-x^2}}} .
Geometric series with a first term {{{b}}},
and common ratio {{{r}}} such that {{{abs(r)<1}}}
converge to {{{b/(1-r)}}} .
Since {{{abs(x)<1}}} means {{{abs(x^3)<abs(x)<1}}},
those 3 series converge to the sums shown above.
Since each series converges, so does its sum.
{{{f(x)=S[1]+S[2]-S[3]=(1-x-x^2)/(1-x^3)}}} .