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Category: Relation and Functions

let-f-x-n-1-sin-nx-n-3-1-study-the-convergence-of-this-serie-2-prove-that-0-pi-f-x-dx-2-n-1-1-2n-1-4-3-prove-that-x-R-f-x-n-1-cos-nx-n-2-4-p

Question Number 37333 by math khazana by abdo last updated on 12/Jun/18 $${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{3}} } \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}=\mathrm{2}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{4}} }…

let-D-D-0-1-and-f-z-n-0-a-n-z-n-is-a-holomorphe-function-f-x-lt-1-1-z-prove-that-a-n-n-1-1-1-n-n-n-1-e-

Question Number 37294 by math khazana by abdo last updated on 11/Jun/18 $${let}\:{D}\:={D}\left(\mathrm{0},\mathrm{1}\right)\:{and}\:{f}\left({z}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{a}_{{n}} {z}^{{n}} \:{is}\:{a}\:{holomorphe} \\ $$$${function}\:/\:\:\mid{f}\left({x}\right)\mid<\:\:\frac{\mathrm{1}}{\mathrm{1}−\mid{z}\mid}\:\:{prove}\:{that} \\ $$$$\mid{a}_{{n}} \mid\leqslant\:\left({n}+\mathrm{1}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)^{{n}} \leqslant\left({n}+\mathrm{1}\right){e}. \\ $$…

let-f-x-x-1-x-2-x-4-1-find-f-n-x-2-calculate-f-n-0-3-developp-f-at-integr-serie-

Question Number 37282 by abdo.msup.com last updated on 11/Jun/18 $${let}\:{f}\left({x}\right)=\frac{{x}}{\mathrm{1}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right){developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$ Commented by abdo.msup.com…

let-f-x-1-1-x-n-with-n-integr-1-find-f-x-and-f-x-2-find-the-poles-of-f-3-calculate-f-n-0-4-developp-f-at-integr-serie-

Question Number 37277 by abdo.msup.com last updated on 11/Jun/18 $${let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{{n}} }\:\:{with}\:{n}\:{integr} \\ $$$$\left.\mathrm{1}\right){find}\:{f}^{'} \left({x}\right)\:{and}\:{f}^{''} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{poles}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right){calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{4}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$ Commented…