Evaluate,
lim(√n/√n^2+ √n/√(n+3)^2+...√n/√(7n- 3)^2
n→∞
limn→∞(nn2+n(n+3)2+⋯+n(7n−3)2)=limn→∞(1n+n(n+3)2+⋯ +n(7n−3)2)=limn→∞(1n+1n(1+1n)2+⋯ +1n(7−3n)2)=0+0+⋯+0=0\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}n→∞lim(n2n+(n+3)2n+⋯+(7n−3)2n)=n→∞lim(n1+(n+3)2n+⋯+(7n−3)2n)=n→∞lim(n1+n(1+n1)21+⋯+n(7−n3)21)=0+0+⋯+0=0
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