Question Number 68238 by mathmax by abdo last updated on 07/Sep/19 $${let}\:\:{f}\left({x}\right)=\left({x}^{\mathrm{2}} −\mathrm{3}{x}\right){arctan}\left(\mathrm{2}{x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$…
Question Number 2675 by prakash jain last updated on 24/Nov/15 $$\mathrm{Bases}\:\mathrm{on}\:\mathrm{suggestion}\:\mathrm{from}\:\mathrm{Filup}\:\mathrm{and}\:\mathrm{some} \\ $$$$\mathrm{discussion}\:\mathrm{on}\:\mathrm{that}\:\mathrm{I}\:\mathrm{am}\:\mathrm{suggesting}\:\mathrm{that}\:\mathrm{we} \\ $$$$\mathrm{sequence},\:\mathrm{series}\:\mathrm{and}\:\mathrm{related}\:\mathrm{function}\:\mathrm{as}\:\mathrm{a} \\ $$$$\mathrm{topic}\:\mathrm{for}\:\mathrm{this}\:\mathrm{month}. \\ $$$$\zeta\left({x}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{n}^{−{x}} ,\:{x}\in\mathbb{R},\:{x}>\mathrm{1} \\ $$$$\mathrm{Show}\:\mathrm{that} \\…
Question Number 68206 by turbo msup by abdo last updated on 07/Sep/19 $${find}\:{S}\left(\theta\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{sin}^{\mathrm{3}} \left({n}\theta\right)}{{n}!} \\ $$ Answered by Smail last updated on 07/Sep/19…
Question Number 2644 by 123456 last updated on 24/Nov/15 $${a}_{{n}+\mathrm{1}} =\frac{{a}_{{n}} }{{n}}+{n} \\ $$$${a}_{\mathrm{1}} =\mathrm{1} \\ $$$${a}_{{n}} =???\:{n}\in\mathbb{N}^{\ast} \\ $$ Commented by RasheedAhmad last updated…
Question Number 2604 by 123456 last updated on 23/Nov/15 $${f}\left({x},{y}\right)={f}\left({x}−\mathrm{1},{y}−{x}\right)+\mathrm{1} \\ $$$${f}\left({x},{y}\right)={ye}^{{x}} ,{x}\leqslant\mathrm{0} \\ $$$${f}\left(\mathrm{5},\mathrm{6}\right)=? \\ $$ Answered by Yozzis last updated on 23/Nov/15 $${f}\left(\mathrm{5},\mathrm{6}\right)={f}\left(\mathrm{4},\mathrm{1}\right)+\mathrm{1}…
Question Number 68129 by mathmax by abdo last updated on 05/Sep/19 $${prove}\:{that}\:\pi{cotan}\left(\alpha\pi\right)=\frac{\mathrm{1}}{\alpha}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{2}\alpha}{\alpha^{\mathrm{2}} −{n}^{\mathrm{2}} } \\ $$$${with}\:\alpha\:\in{R}−{Z}\:\:. \\ $$$${prove}\:{also}\:{that}\:\:\:{for}\:{t}\neq\mathrm{0} \\ $$$${cotan}\left({t}\right)\:=\frac{\mathrm{1}}{{t}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{2}{t}}{{t}^{\mathrm{2}} −{n}^{\mathrm{2}}…
Question Number 68036 by mathmax by abdo last updated on 03/Sep/19 $${let}\:{f}\left({x}\right)\:={cos}\left(\alpha{x}\right)\:\:,\mathrm{2}\pi\:{periodic}\:\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}. \\ $$$$\alpha\:\in\:{R}−{Z} \\ $$ Commented by mathmax by abdo last updated on 05/Sep/19…
Question Number 68034 by mathmax by abdo last updated on 03/Sep/19 $${let}\:{f}\left({x}\right)\:={e}^{−{x}} \:\:,\:\:\mathrm{2}\pi\:\:{periodic}\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}. \\ $$ Commented by mathmax by abdo last updated on 07/Sep/19 $${f}\left({x}\right)\:=\sum_{{n}=−\infty}…
Question Number 68035 by mathmax by abdo last updated on 03/Sep/19 $${let}\:{f}\left({x}\right)\:={e}^{−{i}\alpha{x}} \:\:\:\:,\mathrm{2}\pi\:\:{periodic}\:\:.{developp}\:{f}\:{at}\:{fourier}\:{serie}. \\ $$ Commented by mathmax by abdo last updated on 08/Sep/19 $${f}\left({x}\right)\:=\sum_{{n}=−\infty}…
Question Number 68019 by mathmax by abdo last updated on 03/Sep/19 $${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} +\mathrm{1}} {e}^{−\mathrm{2}{t}} {sin}\left({xt}\right){dt} \\ $$$${determine}\:{F}\:^{'} \left({x}\right)\:{and}\:{calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{F}\left({x}\right). \\ $$$$ \\ $$ Commented…