Monday, 19 August 2013

sequences and series - Calculate limntoinfty(n+1)frac1sqrtn





Calculate the following limit: lim




I have tried to use the squeeze theorem and other convergence tests but all failed.



Please, any help?


Answer



Note that




1\le(n+1)^{1/\sqrt n}\le(2n)^{1/\sqrt n}=2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2



If we take 2^{1/x} and x^{1/x}\to1 as x\to\infty for granted, then



2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2\to1\cdot1^2=1



and the Squeeze Theorem does the rest.


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