Question

How do you know if the series `(1+n+n^2)/(sqrt(1+(n^2)+n^6))` converges or diverges for (n=1 , ∞) ?

Answer 1

`((1+n+n^2)/sqrt(1+n^2+n^6))^2 = (1+2n+3n^2+2n^3+n^4)/(1+n^2+n^6)`

`=1/n^2*(n^2+2n^3+3n^4+2n^5+n^6)/(1+n^2+n^6)`

`=1/n^2*((1+n^2+n^6)+(2n^3+3n^4+2n^5-1))/(1+n^2+n^6)`

`=1/n^2*(1+(2n^3+3n^4+2n^5-1)/(1+n^2+n^6))`

`>1/n^2` for all `n >= 1`

So `(1+n+n^2)/sqrt(1+n^2+n^6) > 1/n`

And `sum_(n=1)^oo (1+n+n^2)/sqrt(1+n^2+n^6) > sum_(n=1)^oo 1/n` diverges.