1.

The sum (2^(1))/(4^(1) - 1) + (2^(2))/(4^(2) - 1) + (2^(4))/(4^(4) - 1) + (2^(8))/(4^(8) - 1) +... oo is equal to

Answer»


SOLUTION :SUM `=sum_(k=0)^(OO)(2^(2^k))/(4^(2^(k))-1) = sum((2^(2^(k)) +1)/(4^(2^(k))-1)-(1)/(4^(2^(k))-1))`
`sum((1)/(2^(2^(k))-1)-(1)/(4^(2^(k))-1))=sum((1)/(2^(2(2^(k-1))))-(1)/(4^(2^(k))-1))`
`= sum_(k=0)^(oo)((1)/(4^(2^(k))-1)-(1)/(4^(2^(k))-1))`
`= ((1)/(4^(2^(1))-1)-(1)/(4^(2^(0))-1))+((1)/(4^(2^(0))-1)-(1)/(4^(2^())-1))+((1)/(4^(2^(1))-1)-(1)/(4^(2^(2))-1))`.....{X}+{-x}={{:(0 if x in I), (1 if x !in I):}`
`= (1)/(4^(-2^(1))-1) = (1)/(2-1) = 1`


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