Question

Investigate convergence or divergence of the following sequence with justification:

(√n2 + 1)(√n).

Answer 1

$\sum _{n=1}^{\infty \:}\sqrt{n^2+1}\sqrt{n}$

Applying series convergence test

$\mathrm{If\:}\lim _{n\to \infty }a_n\ne 0\mathrm{\:then\:}\sum a_n\mathrm{\:diverges}$

$\lim _{n\to \infty \:}\left(\sqrt{n^2+1}\sqrt{n}\right)$

$\lim _{n\to \infty \:}\left(\left(\sqrt{n^2+1}\right)\cdot \lim _{n\to \infty \:}\left(\sqrt{n}\right)\right)$

$\infty \cdot \infty \:$

$\infty$

$\sum _{n=1}^{\infty \:}\sqrt{n^2+1}\sqrt{n}\quad \mathrm{diverges}$