Sequences always blow your mind up

$\sum _{ k=0 }^{ \infty }{ \frac { k }{ k^{ 4 }+k^{ 2 }+1 } }$ = $\frac { 1 }{ 2}\sum _{ k=0 }^{ \infty }{ \frac { 2k }{(k^2 + k + 1)(k^2 - k + 1)} }$

$\frac { 1 }{ 2}\sum _{ k=0 }^{ \infty }{ \frac { (k^2 + k + 1)-(k^2 - k + 1) }{(k^2 + k + 1)(k^2 - k + 1)} }$

All terms cancel out each other in sigma except first term which is equal to 1.

so, $\boxed{x=2}$

Just break k^4 + k^2 + 1 as :- (k^2 + k + 1)(k^2 - k + 1) Then seperate these two terms in form of positive and negative term then apply summation. You will find that all terms will cancel each other out and you will get the answer as 2....

AS SIMPLE AS THAT :)