Question Number 33818 by rahul 19 last updated on 25/Apr/18 $$\boldsymbol{{L}}{et}\:{f}:\boldsymbol{{R}}\:\rightarrow\:\left[\:\mathrm{1},\:\infty\right)\:{be}\:{defined}\:{as}\: \\ $$$${f}\left({x}\right)\:=\:\mathrm{log}_{\mathrm{10}} \:\left(\sqrt{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}{x}+\boldsymbol{{k}}+\mathrm{1}}\:+\mathrm{10}\:\right). \\ $$$$\boldsymbol{{I}}{f}\:{f}\left({x}\right)\:{is}\:\boldsymbol{{surjective}}\:,\:{then}\:{find} \\ $$$${the}\:{value}\:{of}\:\boldsymbol{{k}}\:? \\ $$ Answered by MJS last…
Question Number 33815 by rahul 19 last updated on 25/Apr/18 $$\boldsymbol{{L}}{et}\:{f}:{D}\:\rightarrow\:\boldsymbol{{R}}\:{be}\:{defined}\:{as}\: \\ $$$${f}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+{a}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{3}{a}}\:{where}\:{D}\:{and}\:{R} \\ $$$${denote}\:{the}\:{domain}\:{of}\:\boldsymbol{{f}}\:{and}\:{the}\:{set} \\ $$$${of}\:{all}\:{real}\:{numbers}\:{respectively}. \\ $$$${If}\:{f}\:{is}\:''\:{surjective}\:''\:\:{mapping}\:{then} \\ $$$${the}\:{range}\:{of}\:\boldsymbol{{a}}\:{is}\:? \\ $$$$\left.{a}\right)\:\mathrm{0}\leqslant{a}\leqslant\mathrm{1}…
Question Number 164874 by cortano1 last updated on 22/Jan/22 Answered by mahdipoor last updated on 22/Jan/22 $${get}\:{x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}={t}\:\Rightarrow\:{x}^{\mathrm{2}} +\mathrm{1}={t}^{\mathrm{2}} +{x}^{\mathrm{2}} −\mathrm{2}{xt} \\ $$$$\Rightarrow\:\frac{{t}^{\mathrm{2}} −\mathrm{1}}{\mathrm{2}{t}}={x} \\…
Question Number 33733 by rahul 19 last updated on 22/Apr/18 $${Solve}\:: \\ $$$$\left({x}−\mathrm{2}\right)\:×\:\left[{x}\right]\:=\:\left\{{x}\right\}\:−\mathrm{1}\:. \\ $$$$\bullet\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$$$\bullet\:\left\{.\right\}=\:{fractional}\:{part}\:\:{function}. \\ $$ Commented by MJS last updated on…
Question Number 33719 by prof Abdo imad last updated on 22/Apr/18 $${simplify}\:{S}_{{n}} \left({x}\right)\:=\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{4}} \right)….\left(\mathrm{1}+{x}^{\mathrm{2}^{{n}} } \right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \left({x}\right)\:{if}\:\mid{x}\mid<\mathrm{1}\:. \\ $$ Commented by…
Question Number 33717 by prof Abdo imad last updated on 22/Apr/18 $${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{artan}\left(\:\frac{\sqrt{{n}+\mathrm{1}}\:−\sqrt{{n}}}{\mathrm{1}+\sqrt{{n}^{\mathrm{2}} +{n}}}\:\right) \\ $$ Commented by prof Abdo imad last updated on…
Question Number 33718 by prof Abdo imad last updated on 22/Apr/18 $${find}\:\:\sum_{{n}=\mathrm{1}} ^{+\infty} {arctan}\left(\:\frac{{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)\:−{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)}{\mathrm{1}+\left(\mathrm{1}\:+\frac{\mathrm{1}}{{n}}\right)\left(\mathrm{1}+\frac{\mathrm{1}}{\left.{n}+\mathrm{1}\right)}\right.}\right) \\ $$ Commented by prof Abdo imad last updated on 26/Apr/18…
Question Number 33716 by prof Abdo imad last updated on 22/Apr/18 $${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:{arctan}\left(\:\frac{{e}^{{n}+\mathrm{1}} \:\:−{e}^{{n}} }{\mathrm{1}+{e}^{\mathrm{2}{n}+\mathrm{1}} }\right)\:. \\ $$ Commented by prof Abdo imad last…
Question Number 33713 by abdo imad last updated on 22/Apr/18 $${find}\:{tbe}\:{value}\:{of}\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\frac{{n}^{\mathrm{2}} −{n}+\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{2}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$ Commented by prof Abdo imad last updated…
Question Number 33710 by math khazana by abdo last updated on 22/Apr/18 $$\left.\mathrm{1}\right)\:{find}\:{the}\:{radius}\:{of}\:{convergence}?{for} \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{x}^{{n}} }{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}\:{and}\:{calculate}\:{its}\:{sum} \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}\:\mathrm{2}^{{n}}…