`sum_(r=1)^oo((4r+5)5^(-r))/(r(5r+5))`

Options

A. `(1)/(5)`
B. `(2)/(5)`
C. `(1)/(25)`
D. `(2)/(25)`

Answer

A

Explanation

`sum_(r=1)^(oo)((4r+5)5^(-r))/(r(5r+5))= underset(n to oo)(lim)sum_(r=1)^(n)(((5r+5)-r)/(r(5r+5)))*(1)/(5^(r))`
`=underset(n to oo)(lim)sum_(r=1)^(n)((1)/(r)-(1)/(5r+5))(1)/(5^(r))`
`=underset(n to oo)(lim)sum_(r=1)^(n)((1)/(r*5^(-r))-(1)/((r+1)5^(r+1)))`
`=underset(n to oo)(lim)sum_(r=1)^(n)((1)/(5)-(1)/((n+1)5^(r+1)))=(1)/(5)-0=(1)/(5)`.