Tuesday, 4 February 2014

real analysis - limntoinftyfracsqrtn(sqrt1+sqrt2+...+sqrtn)n2



How do I find the following limit?
lim
Can limit be find by Riemann sums?

\lim_{n\to \infty}\sum_{k=1}^{n}f(C_k)\Delta{x} = \int_{a}^{b}f(x)\,dx
I'm not sure what f(C_k) is.


Answer



Hint:
\lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + ... + \sqrt{n}}{\sqrt{n}}\frac1n= \lim_{n \to \infty} \left(\sqrt{\frac1n} + \sqrt{\frac2n} + \sqrt{\frac3n} +\cdots+\sqrt{\frac{n}{ n}} \right) \frac1n
f(C_k)=\sqrt{\dfrac{k}{n}} and \Delta x=\dfrac1n .


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