[tex]a_{n}=1+ \frac{2}{3} + \frac{3}{9} + \frac{4}{27} + ...+\frac{n+1}{3^n}
[/tex]
[tex]a_{n}=(1+ \frac{1}{3} + \frac{1}{9} +...+ \frac{1}{3^n} )+(\frac{1}{3} + \frac{1}{9} +...+ \frac{1}{3^n} )+(\frac{1}{9} +...+ \frac{1}{3^n} )+...[/tex]
[tex]a_{n}=1 \cdot \frac{ (\frac{1}{3})^{n+1}-1 }{ \frac{1}{3} -1} + \frac{1}{3} \cdot \frac{ (\frac{1}{3})^{n}-1 }{ \frac{1}{3} -1}+\frac{1}{9} \cdot \frac{ (\frac{1}{3})^{n-1}-1 }{ \frac{1}{3} -1}+...[/tex]
[tex]a_{n}= \frac{3}{2} \cdot (1+ \frac{1}{3} + \frac{1}{9} +...)[/tex]
[tex] \lim_{n \to \infty} a_n = \frac{9}{4} [/tex]
