The sum to 10   terms of the series 1 1 + 1 2 + 1 4 + 2 1 + 2 2 + 2 4 + 3 1 + 3 2 + 3 4 + … is :-

We have,

$\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots$

Here,

$t_n=\frac{n}{1+n^2+n^4}$

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

So,

$t_1=\frac{1}{2}\left(1-\frac{1}{3}\right)$

$t_2=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{7}\right)$

$$\begin{gathered} ⋮ ⋮ ⋮ \\ ⋮ ⋮ ⋮ \\ ⋮ ⋮ ⋮ \end{gathered}$$

$t_{10}=\frac{1}{2}\left(\frac{1}{91}-\frac{1}{111}\right)$

Now,

$S_{10}=t_1+t_2+t_3+.....+t_{10}$

$\Rightarrow S_{10}=\frac{1}{2}\left(1-\frac{1}{111}\right)$

$\Rightarrow S_{10}=\frac{55}{111}$