I am trying to sum the following sequence to calculate mean recurrence time $\mu_{00}$ in markov chain:
$\mu_{00}$ = 1.$\frac{1}{2}$+2.$\frac{1}{2}$.$\frac{3}{4}$+3.0+4.$\frac{1}{2}$.$\frac{1}{4}$.1.$\frac{3}{4}$+5.0+6.$\frac{1}{2}$.$\frac{1}{4}$.1.$\frac{1}{4}$.1.$\frac{3}{4}$+7.0+8.$\frac{1}{2}$.$\frac{1}{4}$.1.$\frac{1}{4}$.1.$\frac{1}{4}$.1.$\frac{3}{4}$..........
I have managed to solve this up to as under:-
$\mu_{00}$ =$\frac{1}{2}$ + $\frac{3}{4}$+$\frac{1}{2}$.$\frac{3}{4}$[$\frac{4}{4}$+$\frac{6}{4^{2}}$+$\frac{8}{4^{3}}$+$\frac{10}{4^{4}}$.........]
= $\frac{5}{4}$ +$\frac{3}{4}$[$\frac{2}{4}$+$\frac{3}{4^{2}}$+$\frac{4}{4^{3}}$+$\frac{5}{4^{4}}$.........]
= $\frac{5}{4}$ +$\frac{3}{4}${[$\frac{1}{4^{0}}$+$\frac{2}{4^{1}}$+$\frac{3}{4^{2}}$+$\frac{4}{4^{3}}$+$\frac{5}{4^{4}}$.........]-1}
I am unable to sum the series given in the bracket... it is neither an arithmatic or a geometric progression. Request help.