<s>[INST] Solution of this summation: $\sum_{i=1}^{10} {\frac{i}{i^4+i^2+1}}$ The summation in question:
$$\sum_{i=1}^{10} {\frac{i}{i^4+i^2+1}}$$
I have been able to factorize $i^4+i^2+1$ as $(i^2+i+1)(i^2-i+1)$ but I doubt this will help.
What is the solution?
 [/INST] Writting $$\sum^{10}_{i=1}\frac{i}{i^4+i^2+1} = \frac{1}{2}\sum^{10}_{i=1}\bigg[\frac{(i^2+i+1)-(i^2-i+1)}{(i^2+i+1)(i^2-i+1)}\bigg]$$
$$ = \frac{1}{2}\sum^{10}_{i=1}\bigg[\frac{1}{i^2-i+1}-\frac{1}{i^2+i+1}\bigg]$$
Now Using Telescopic Sum (expanding summation), Then we get 
$$ =\frac{1}{2}\bigg[\frac{1}{1^2-1+1}-\frac{1}{10^2+10+1}\bigg]= \frac{1}{2}\left(1-\frac{1}{111}\right) = \frac{55}{111}$$
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