is it just limits?

Calculus Level 4

$\lim_{n \to \infty} \left(\frac 1{\sqrt n \sqrt {n+1}}+\frac 1{\sqrt n \sqrt {n+2}}+\cdots+\frac 1{\sqrt n \sqrt {n+n}}\right)$

Find the value of the closed form of the above limit to 2 decimal places.

The answer is 0.83.

1 solution

Brian Moehring
Feb 2, 2017

Relevant wiki: Riemann Sums

$\sum_{k=1}^n \frac{1}{\sqrt{n}\sqrt{n+k}} = \sum_{k=1}^n \frac{1}{\sqrt{1+\frac{k}{n}}} \cdot \frac{1}{n} \xrightarrow{n\rightarrow \infty} \int_0^1 \frac{1}{\sqrt{1+x}}\,dx = 2\sqrt{1+x}\Big|_0^1 = 2\sqrt{2} - 2 \approx \boxed{0.83}$

is it just limits?

The answer is 0.83.

1 solution

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