(1/(1+1^2 +1^4 ))+(2/(1+2^2 +2^4 ))+(3/(1+3^2 +3^4 ))+...+((50)/(1+50^2 +50^4 ))=?


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Question Number 125864 by MathSh last updated on 14/Dec/20

$\frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + . . . + \frac{50}{1 + 50^{2} + 50^{4}} = ?$
	Answered by Dwaipayan Shikari last updated on 14/Dec/20

	$\underset{n = 1}{\sum^{n}} \frac{n}{n^{4} + n^{2} + 1} = \underset{n = 1}{\sum^{n}} \frac{n}{(n^{2} + 1 - n) (n^{2} + n + 1)} = \frac{1}{2} \sum^{n} \frac{1}{n^{2} - n + 1} - \frac{1}{n^{2} + n + 1}$ $= \frac{1}{2} (\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{7} + \frac{1}{7} - . . . - \frac{1}{n^{2} + n + 1})$ $= \frac{1}{2} (1 - \frac{1}{n^{2} + n + 1}) = \frac{n^{2} + n}{2 (n^{2} + n + 1)}$ $n = 50 t h e n S u m i s \frac{1}{2} (1 - \frac{1}{2551}) = \frac{1275}{2551}$
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