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Prove that sum(r=1)^(n) tan^(-1) ((2^(r...

Prove that `sum_(r=1)^(n) tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = tan^(-1) (2^(n)) - (pi)/(4)`

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`underset(r =1) overset(n) sum tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = underset( r=1)overset(n) sum tan^(-1) ((2^(r -1))/(1 + 2^(r).2^(r-1)))`
`= underset(r =1) overset(n)sum tan^(-1) ((2^(r) -2^(r-1))/(1 + 2^(r).2^(r -1)))`
`= underset(r=1)overset(n) sum [tan^(-1) (2^(r)) - tan^(-1) (2^(r -1))]`
`= tan^(-1) (2^(n)) - tan^(-1) (1)`
`= tan^(-1) (2^(n)) - (pi)/(4)`
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