Question

Find the sum of n terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+.....$

Answer 1

$\Rightarrow u_n=\frac{n}{1+n^2+n^4}=\frac{1}{2}(\frac{1}{1-n+n^2}-\frac{1}{1+n+n^2})$

$u_n=\frac{1}{2}(\frac{1}{1-n+n^2}-\frac{1}{1+n+n^2})$

$u_{n-1}=\frac{1}{2}(\frac{1}{2-3n+n^2}-\frac{1}{1-n+n^2})$

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$u_1=\frac{1}{2}(\frac{1}{1}-\frac{1}{3})$

$S_n=\frac{1}{2}(1-\frac{1}{1+n+n^2})$