Solveeit Logo
Login

Question

Mathematics Question on limits and derivatives

lim⁡n→∞(12+22+....+n2)nn(n+1)(n+10)(n+100)=\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

33

B

1/31/3

C

2/32/3

D

∞ \infty

Answer

1/31/3

Explanation

Solution

lim⁡n→∞n(n+1)(2n+1)nn6(n+1)(n+10)(n+100)\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)}
=lim⁡n→∞(2+1n)lim⁡n→∞nn6(1+10n)(1+100n)=26=13= \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}