let-f-x-n-1-1-n-cos-n-x-sin-nx-1-prove-the-convergence-of-this-serie-2-prove-that-f-is-C-2-on-R-kpi-k-Z-and-calculate-f-x-3-give-a-exprrssion-of-f-

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Question Number 36744 by prof Abdo imad last updated on 05/Jun/18

let f(x)=Σ_(n=1) ^∞ (1/n) cos^n (x)sin(nx) 1)prove the convergence of this serie 2)prove that f is C^2 on R −{kπ,k∈Z}and calculate f^′ (x) 3) give a exprrssion of f.

$${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}}\:{cos}^{{n}} \left({x}\right){sin}\left({nx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{2}} \:{on}\:{R}\:−\left\{{k}\pi,{k}\in{Z}\right\}{and} \\ $$$${calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{a}\:{exprrssion}\:{of}\:{f}. \\ $$

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let-f-x-n-1-1-n-cos-n-x-sin-nx-1-prove-the-convergence-of-this-serie-2-prove-that-f-is-C-2-on-R-kpi-k-Z-and-calculate-f-x-3-give-a-exprrssion-of-f-

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