2011/027) obtain the sum of

1/(x+1) + 2/(x^2+1) + 4/(x^4+1) + (2^2n/(x^2n + 1)

we realise that 1/(x+1) - 1/(x-1) = -2 /(x^2-1)

so add and subtract 1/(x-1) to get

1/(x-1) + (1/(x+1) - 1/(x- 1) + 2/(x^2+1) + 4/(x^4+1) +.... (2^2n/(x^2n + 1)))
= 1/(x-1) + (-2/(x^2-1)+ 2/(x^2+1) + 4/(x^4+1) + (2^2n/(x^2n + 1)))
= 1/(x-1) + (-4/(x^4-1)+ ... + (2^2n/(x^2n + 1)))

applying repeatedly we get 2^(4n)/(x^4n+1) + 1/(x-1)