Question

$\displaystyle\lim_{n\to\infty} \frac{\left(1^{2} +2^{2} + ....+n^{2}\right) \sqrt[n]{n}}{ \left(n+1\right)\left(n+10\right)\left(n+100\right)} = $

• A. $3$
• B. $1/3$
• C. $2/3$
• D. $ \infty$

Answer 1

Answer: (B)

$\displaystyle\lim_{n\to\infty} \frac{n\left(n+1\right)\left(2n+1\right)\sqrt[n]{n}}{6\left(n+1\right)\left(n+10\right)\left(n+100\right)}$
$ = \displaystyle\lim _{n\to \infty } \frac{\left(2+\frac{1}{n}\right) \displaystyle\lim _{n\to \infty } \sqrt[n]{n}}{6\left(1+\frac{10}{n}\right)\left(1+\frac{100}{n}\right)} = \frac{2}{6} = \frac{1}{3}$