Question

Evaluate,

lim(√n/√n^2+ √n/√(n+3)^2+...√n/√(7n- 3)^2

n→∞

Answer 1

$\displaystyle \begin{aligned} \lim_{n \to \infty} \left(\frac{\sqrt{n}}{\sqrt{n^2}} + \frac{\sqrt{n}}{\sqrt{(n + 3)^2}} + \cdots + \frac{\sqrt{n}}{\sqrt{(7n - 3)^2}}\right) &= \lim_{n \to \infty} \left(\sqrt{\frac{1}{n}} + \sqrt{\frac{n}{(n + 3)^2}} + \cdots \right. \\&\left.+\sqrt{\frac{n}{(7n - 3)^2}}\right) \\&= \lim_{n \to \infty} \left(\sqrt{\frac{1}{n}} + \sqrt{\frac{1}{n\left(1 + \frac{1}{n}\right)^2}} + \cdots \right. \\&\left. +\sqrt{\frac{1}{n\left(7 - \frac{3}{n}\right)^2}}\right) \\&= 0 + 0 + \cdots + 0 = 0 \end{aligned}$