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Category: Relation and Functions

calculate-n-1-1-n-1-n-meant-the-floor-

Question Number 97795 by abdomathmax last updated on 09/Jun/20 $$\mathrm{calculate}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} }{\left[\sqrt{\mathrm{n}}\right]} \\ $$$$\left[..\right]\:\mathrm{meant}\:\mathrm{the}\:\mathrm{floor} \\ $$ Answered by bobhans last updated on 10/Jun/20 $$\lfloor\sqrt{\mathrm{n}}\:\rfloor\:=\:\mathrm{m}\:\in\mathbb{N}…

Determine-all-function-f-R-0-1-R-satisfying-the-functional-relation-f-x-f-1-1-x-2-1-2x-x-1-x-x-0-x-1-

Question Number 97648 by bobhans last updated on 09/Jun/20 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{function}\:\mathrm{f}:\mathrm{R}/\left\{\mathrm{0},\mathrm{1}\right\}\rightarrow\mathrm{R} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{functional}\:\mathrm{relation} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{1}−\mathrm{x}}\right)\:=\:\frac{\mathrm{2}\left(\mathrm{1}−\mathrm{2x}\right)}{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)}\:,\:\mathrm{x}\neq\mathrm{0},\:\mathrm{x}\neq\mathrm{1} \\ $$ Commented by bemath last updated on 09/Jun/20 $$\mathrm{nice}\:\mathrm{question} \\…

let-f-x-arctan-x-2-3-1-calculate-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-3-calculate-0-1-f-x-dx-

Question Number 97622 by mathmax by abdo last updated on 08/Jun/20 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\mathrm{arctan}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{3}\right) \\ $$$$\left.\mathrm{1}\right)\:\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$…

let-u-n-0-1-x-n-sin-pix-dx-1-prove-that-u-n-converges-2-prove-that-u-n-0-pi-sint-t-dt-

Question Number 32042 by abdo imad last updated on 18/Mar/18 $${let}\:{u}_{{n}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{x}^{{n}} \:{sin}\left(\pi{x}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Sigma\:{u}_{{n}} \:{converges} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\Sigma\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sint}}{{t}}{dt}\:. \\ $$…