Monday, 25 April 2016

integration - Find limnrightarrowinftyfrac1nintinfty1fracmathrmdxx2log(1+fracxn)



Find:



lim



The sequence \frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ \frac{x}{n})}}{\frac{x}{n}}} converges pointwise to \frac{1}{x^3}. So if we could apply Lebesgue's Dominated Convergence Theorem, we have:




\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{\mathrm dx}{x^2 \log{(1+ \frac{x}{n})}}=\lim_{n \rightarrow \infty} \int_{1}^{\infty} \frac{\mathrm dx}{x^3}=\frac{1}{2}



I have a problem with finding a majorant. Could someone give me a hint?


Answer



I showed in THIS ANSWER, using only Bernoulli's Inequality the sequence \left(1+\frac xn\right)^n is monotonically increasing for x>-n.



Then, we can see that for x\ge 1 and n\ge1, the sequence f_n(x) given by



f_n(x)=n\log\left(1+\frac xn\right)




is also monotonically increasing. Therefore, a suitable dominating function is provided simply by the inequality



\frac{1}{n\log\left(1+\frac xn\right)}\le \frac{1}{\log(1+x)}\le \frac{1}{\log(2)}



Therefore, we have



\frac{1}{nx^2\log\left(1+\frac xn\right)}\le \frac{1}{x^2\log(2)}



Using the dominated convergence theorem, we can assert that




\begin{align} \lim_{n\to \infty}\int_1^\infty \frac{1}{nx^2\log\left(1+\frac xn\right)}\, dx&=\int_1^\infty \lim_{n\to \infty}\left(\frac{1}{nx^2\log\left(1+\frac xn\right)}\right)\,dx\\\\ &=\int_1^\infty\frac{1}{x^3}\,dx\\\\ &=\frac12. \end{align}


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