An infinite series

Algebra Level 3

$\dfrac{1}{1+1^2 + 1^4} +\dfrac{2}{1+2^2 +2^4}+\dfrac{3}{1+3^2 +3^4}+ \ldots$

1 solution

Sandeep Rathod
Nov 16, 2014

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

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

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

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

Since $n$ tends to $\infty$ the answer becomes $\boxed{\frac{1}{2}}$

Edited for clarity.

Krishna Ar - 6 years, 6 months ago

OK Thank you

sandeep Rathod - 6 years, 6 months ago

i did the same way!!

Aareyan Manzoor - 6 years, 5 months ago

I did in the same way

Sudhir Aripirala - 5 years, 11 months ago

nice solution

Faheem Mubarak - 1 year, 2 months ago