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Evaluate the sum of the first n terms of the series 1/(1+1²+1⁴) + 2/(1+2²+2⁴) + 3/(1+3²+3⁴) +...

1. (n²+n-1)/(2(n²+n+1))
2. (n²+n)/(2(n²+n+1))
3. (n²-n+1)/(n²+n+1)
4. (n²-n)/(2(n²+n+1))

Correct answer: (n²+n)/(2(n²+n+1))

Solution

k/(1+k^2+k^4)=(1/2)[1/(k^2-k+1)-1/(k^2+k+1)], a telescoping sum. The total is (1/2)[1 - 1/(n^2+n+1)] = (n^2+n)/(2(n^2+n+1)).

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