# Infinite sum of a series

Find the value of $$\sum_{r=1} ^\infty \frac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)}$$ I tried converting the general term into partial fractions but the $$(r^2+1)^2$$ in the numerator is creating problems to do the sum. Please can somebody explain how can I compute the infinite sum

Write the numerator as $(r^4+r^2+1)+(r^3+r^2)$ to get $$\frac 1{r^2+r}+\frac r{r^4+r^2+1}$$

And go from there.

Hint:

$r^4+r^2+1=(r^2+r+1)(r^2-r+1)$

Write denominator as product of $r(r^2-r+1)$ and $(r+1)(r^2+r+1)$

As $f(x)=x(x^2-x+1),f(x+1)=?$

• – lab bhattacharjee Nov 16 '17 at 16:34
• I love this trick, completing the square and arriving at this great substitution. +1 on my end – JohnColtraneisJC Nov 16 '17 at 20:54