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let-f-x-0-e-x-t-sin-xt-dt-with-x-gt-0-1-find-a-explicit-form-for-f-x-2-let-U-n-nf-n-find-lim-n-U-n-and-study-the-convergence-of-U-n-




Question Number 58212 by maxmathsup by imad last updated on 20/Apr/19
let f(x) =∫_0 ^∞   e^(−x[t])  sin(xt)dt   with x>0  1) find a explicit form for f(x)  2) let U_n =nf(n)   find lim_(n→+∞)  U_n    and study the convergence of ΣU_n
$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}\left[{t}\right]} \:{sin}\left({xt}\right){dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{let}\:{U}_{{n}} ={nf}\left({n}\right)\:\:\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{U}_{{n}} \:\:\:{and}\:{study}\:{the}\:{convergence}\:{of}\:\Sigma{U}_{{n}} \\ $$

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