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Question
Sum the following series:
`tan^-1 1/3+tan^-1 2/9+tan^-1 4/33+...+tan^-1 (2^(n-1))/(1+2^(2n-1))`
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Solution
`tan^-1 1/3+tan^-1 2/9+tan^-1 4/33+...+tan^-1 (2^(n-1))/(1+2^(2n-1))`
⇒ `tan^-1((2-1)/(1+2xx1))+tan^-1((4-2)/(1+4xx2))+tan^-1((8+4)/(1+8xx4))+...+tan^-1((2^n-2^n-1)/(1+2^n.2^(n-1))`
⇒ `(tan^-1 2-tan^-1 1)+(tan^-1 4-tan^-1 2)+(tan^-1 8-tan^-1 4)+...+(tan^-1 2^(n-1)-tan^-1 2^(n-2))+(tan^-1 2^n-tan^-1 2(n-1))`
⇒ `tan^-1 2^n-tan^-1 1`
⇒ `tan^-1 2^n -pi/4`
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