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Category: Relation and Functions

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

Question Number 33256 by prof Abdo imad last updated on 14/Apr/18 $${let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\left({x}\right)\:{at}\:{integr}\:{serie}. \\ $$$$ \\ $$…

f-is-a-continue-and-positive-function-on-a-b-with-a-lt-b-let-m-max-x-a-b-f-x-prove-that-lim-n-1-b-a-a-b-f-n-x-dx-1-n-

Question Number 33167 by abdo imad last updated on 11/Apr/18 $${f}\:{is}\:{a}\:{continue}\:{and}\:{positive}\:{function}\:{on}\:\left[{a},{b}\right]\:{with}\:{a}<{b} \\ $$$${let}\:{m}\:={max}_{{x}\in\left[{a},{b}\right]} \:{f}\left({x}\right)\:{prove}\:{that} \\ $$$${lim}_{{n}\rightarrow\infty} \:\:\left(\:\frac{\mathrm{1}}{{b}−{a}}\:\int_{{a}} ^{{b}} \:{f}^{{n}} \left({x}\right){dx}\right)^{\frac{\mathrm{1}}{{n}}} \\ $$ Commented by abdo…

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Question Number 33131 by prof Abdo imad last updated on 11/Apr/18 $$\left.\mathrm{1}\right){find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{e}^{{inx}} }{{n}\left({n}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{{sin}\left({nx}\right)}{{n}\left({n}+\mathrm{1}\right)} \\ $$$${and}\:\sum_{{n}\geqslant\mathrm{1}} \:\:\frac{{cos}\left({nx}\right)}{{n}\left({n}+\mathrm{1}\right)}\:. \\ $$ Commented by…