Answer to Question #206986 in Calculus for Rajkumar

Question

test ∑n=1 ∞​ [✓n^4 +9 - ✓n^4 -9] is convergent or not

Answer 1

$\Sigma_{n=1}^{\infty}(\sqrt{n^4+9}-\sqrt{n^4-9})=\\ =\Sigma_{n=1}^{\infty}\frac{(\sqrt{n^4+9}-\sqrt{n^4-9})(\sqrt{n^4+9}+\sqrt{n^4-9})}{(\sqrt{n^4+9}+\sqrt{n^4-9})}=\\ =\Sigma_{n=1}^{\infty}\frac{18}{\sqrt{n^4+9}+\sqrt{n^4-9}}\\ \frac{18}{\sqrt{n^4+9}+\sqrt{n^4-9}}\leftrightarrow\frac{18}{2n^2}=\frac{9}{n^2}$

$\Sigma_{n=1}^{\infty}\frac{9}{n^2}$ is convergent

$\Sigma_{n=1}^{\infty}(\sqrt{n^4+9}-\sqrt{n^4-9})$ is convergent