Question

If \[ u_n=\frac{\sqrt{2n^2-5n+1}}{4n^3-7n^2+2} \] and \[ v_n=\sqrt{n^2+1}-\sqrt{n^2-1}, \] then

• A. \(\displaystyle\sum_{n=1}^{\infty}u_n\) convergent and \(\displaystyle\sum_{n=1}^{\infty}v_n\) divergent
• B. \(\displaystyle\sum_{n=1}^{\infty}u_n\) and \(\displaystyle\sum_{n=1}^{\infty}v_n\), both converge
• C. \(\displaystyle\sum_{n=1}^{\infty}u_n\) divergent and \(\displaystyle\sum_{n=1}^{\infty}v_n\) convergent
• D. \(\displaystyle\sum_{n=1}^{\infty}u_n\) and \(\displaystyle\sum_{n=1}^{\infty}v_n\), both diverge

Hint:

Use asymptotic comparison: \[ \boxed{ u_n\sim\frac1{n^2}\Rightarrow\text{Convergent}, \qquad v_n\sim\frac1n\Rightarrow\text{Divergent}. } \]

Answer 1

The Correct Option is A