Saturday, December 9, 2017

calculus - Verify limlimitsnrightarrowinftyintinfty0!enxsin(ex),mathrmdx=0.



How to verify lim?



My idea is to use the dominant convergence theorem with f_n(x):= e^{-nx} \sin(e^x) and f(x):=\lim\limits_{n\rightarrow\infty}f_n(x).




\Rightarrow \lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = \int^\infty_0 \! \lim\limits_{n\rightarrow\infty} e^{-nx} \sin(e^x) \, \mathrm{d}x = \int^\infty_0 \!\lim\limits_{n\rightarrow\infty}0\, \mathrm{d}x = 0



Can I use this here?


Answer




How to verify \lim\limits_{n\rightarrow\infty}\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x = 0




You may just observe that

\left|\int^\infty_0 \! e^{-nx} \sin(e^x) \, \mathrm{d}x\right|\leq \int^\infty_0 \! \left|e^{-nx} \right|\, \mathrm{d}x=\frac1n, \qquad (n>0), and then let n \to +\infty.


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