Thursday, 1 January 2015

real analysis - Calculate the limit left(frac1sqrtn2+1+frac1sqrtn2+2+...+frac1sqrtn2+nright)



I have to calculate the following limit:
lim



I believe the limit equals 1, and I think I can prove it with the squeeze theorem, but I don't really know how.




Any help is appreciated, I'd like to receive some hints if possible.



Thanks!


Answer



For every n>0,
\frac{n}{\sqrt{n^2+n}}\le\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\le\frac{n}{\sqrt{n^2+1}}



Can you continue with Squeeze theorem?


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