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


Question and Answers Forum

All Questions Topic List
Algebra Questions
Previous in All Question Next in All Question
Previous in Algebra Next in Algebra
Question Number 12144 by rish@bh last updated on 14/Apr/17

$\frac{1}{1 + 1^{2} + 1^{4}} + \frac{2}{1 + 2^{2} + 2^{4}} + \frac{3}{1 + 3^{2} + 3^{4}} + . . . . . \infty = ?$
	Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 14/Apr/17

	$\frac{n}{n^{4} + n^{2} + 1} = \frac{n}{{(n^{2} + 1)}^{2} - n^{2}} = \frac{n}{(n^{2} + n + 1) (n^{2} - n + 1)} =$ $\frac{1}{2} . \frac{(n^{2} + n + 1) - (n^{2} - n + 1)}{(n^{2} + n + 1) (n^{2} - n + 1)} =$ $\frac{1}{2} (\frac{1}{n^{2} - n + 1} - \frac{1}{n^{2} + n + 1})$ $L H S = \frac{1}{2} [\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{7} + \frac{1}{7} - \frac{1}{13} . . . . . - \frac{1}{n^{2} + n + 1}] =$ $= \frac{1}{2} . (1 - \frac{1}{n^{2} + n + 1}) = \frac{n^{2} + n}{2 (n^{2} + n + 1)}$ $\lim_{x \to\propto} L H S = \frac{1}{2} . ◼$
	Commented by rish@bh last updated on 14/Apr/17

	$Thanks$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum