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sum(r=1)^oo((4r+5)5^(-r))/(r(5r+5))...

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

A

`(1)/(5)`

B

`(2)/(5)`

C

`(1)/(25)`

D

`(2)/(25)`

Text Solution

Verified by Experts

The correct Answer is:
A
`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)`.
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