Show $$\lim_{n \to \infty} \int_0^1 \frac{n+n^k x^k}{(1+\sqrt{x})^n}dx=0$$ for $k=1,2,3,...$ It's clear that the functions converge pointwise to $0$ on $(0,1]$ but I can't seem to find an integrable dominating function. Any hints would be much appreciated.
Measure theory convergence question.
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real-analysis
measure-theory
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0I probably do not understand the problem. Take for example the $n/(1+\sqrt{x})^n$ part. Split the bottom as $(1+\sqrt{x})^{n/2}(1+\sqrt{x})^{n/2}$. The second one is bigger than $1+(n/2)\sqrt{x}$, which is bigger than $(n/2)\sqrt{x}$. So $n/(1+\sqrt{x})^n<2/(\sqrt{x}(1+\sqrt{x})^{n/2})$. – 2012-01-12