Question Number 40893 by abdo.msup.com last updated on 28/Jul/18 $${let}\:{u}_{{k}} =\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{k}} }\right)^{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\Sigma\:{u}_{{k}} {converges} \\ $$$$\left.\mathrm{2}\right){let}\:{f}\left({x}\right)=\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\right)^{{n}−\mathrm{1}} \:{with}\:{x}\geqslant\mathrm{0} \\ $$$${prove}\:{that}\:\forall{p}\in{N} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{p}+\mathrm{1}} \:{u}_{{k}}…
Question Number 40891 by abdo.msup.com last updated on 28/Jul/18 $${let}\:{x}>\mathrm{0}\:{and}\:{y}>\mathrm{0}\:{and} \\ $$$${B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{B}\left({x},{y}\right)={B}\left({y},{x}\right) \\ $$$$\left.\mathrm{2}\right){B}\left({x}+\mathrm{1},{y}\right)=\frac{{x}}{{y}}\:{B}\left({x},{y}+\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right){B}\left({x}+\mathrm{1},{y}\right)=\frac{{x}}{{x}+{y}}{B}\left({x},{y}\right) \\ $$$$\left.\mathrm{4}\right){B}\left({x},{n}+\mathrm{1}\right)=\frac{{n}!}{{x}\left({x}+\mathrm{1}\right)….\left({x}+{n}\right)} \\…
Question Number 40890 by abdo.msup.com last updated on 28/Jul/18 $$\left.\mathrm{1}\right){calculate}\:\int_{\frac{\mathrm{1}}{{n}+\mathrm{1}}} ^{\frac{\mathrm{1}}{{n}}} \left[\frac{\mathrm{1}}{{t}}−\left[\frac{\mathrm{1}}{{t}}\right]\right]{dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left[\frac{\mathrm{1}}{{t}}−\left[\frac{\mathrm{1}}{{t}}\right]\right]{dt}=\mathrm{1}−\gamma \\ $$$$\gamma\:{is}\:{constant}\:{number}\:{of}\:{euler} \\ $$ Commented by maxmathsup by imad…
Question Number 40889 by abdo.msup.com last updated on 28/Jul/18 $${prove}?{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\left(\mathrm{1}−{t}\right)^{{n}} }{{t}}{dt}\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$ Answered by math khazana by abdo…
Question Number 40888 by prof Abdo imad last updated on 28/Jul/18 $$\left.{prove}\:{that}\:\forall\xi\:\in\right]\mathrm{0},\pi\left[\right. \\ $$$$\mid\int_{\xi} ^{\pi} \left(\frac{{sint}}{{t}}\right)^{{n}} {dt}\mid\leqslant\pi\left(\frac{{sin}\xi}{\xi}\right)^{{n}} \:\:{n}\:{integr}\:{natural} \\ $$ Terms of Service Privacy Policy…
Question Number 40886 by prof Abdo imad last updated on 28/Jul/18 $${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{t}^{\mathrm{2}{p}+\mathrm{1}} {ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{4}}\sum_{{k}=\mathrm{1}} ^{{p}} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} } \\ $$ Terms…
Question Number 40887 by prof Abdo imad last updated on 28/Jul/18 $${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{tln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt} \\ $$ Commented by prof Abdo imad last updated on…
Question Number 40884 by prof Abdo imad last updated on 28/Jul/18 $$\left.\mathrm{1}\right)\:{fond}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({t}\right)}{{t}^{\mathrm{4}} −\mathrm{1}}{dt} \\ $$ Answered by maxmathsup…
Question Number 40883 by prof Abdo imad last updated on 28/Jul/18 $${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{p}} }{{e}^{{t}} −\mathrm{1}}{dt}\:{with}\:{p}\in{N}^{\bigstar} \\ $$ Answered by maxmathsup by imad last updated…
Question Number 40870 by math khazana by abdo last updated on 28/Jul/18 $${fnd}\:\:\int\:\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctan}\left({x}−\frac{\mathrm{1}}{{x}}\right){dx}\:. \\ $$ Commented by maxmathsup by imad last updated on 29/Jul/18…