Question Number 26244 by abdo imad last updated on 22/Dec/17 $$\:\left({x}_{{i}} \:\right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \:\:{n}\:{real}\:{number}\:\:{positifs}\:{wish}\:{verfy}\:\:\:\sum_{{i}=\mathrm{1}} ^{{i}={n}} \:{x}_{{i}} =\mathrm{1} \\ $$$${prove}\:{that}\:\:\:\sum_{\mathrm{1}\leqslant{i}\leqslant{n}} {x}_{{i}} ^{\mathrm{2}} \:\:\:\geqslant\:\:\frac{\mathrm{1}}{{n}}\:\:\:. \\ $$ Commented by…
Question Number 26243 by abdo imad last updated on 22/Dec/17 $${let}\:{put}\:{U}_{{n}} \:\:=\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\:\frac{\mathrm{1}}{{ij}}\:\:\:\:{find}\:\:{lim}_{{n}−>\propto} \:\:{U}_{{n}} \\ $$ Commented by abdo imad last updated on 28/Dec/17 $$\:{we}\:{have}\:{the}\:{equality}\:\:\:\:\:\left({x}_{\mathrm{1}}…
Question Number 157314 by cortano last updated on 22/Oct/21 $$\:\:\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{3}\:\mathrm{ln}\:\left(\mathrm{1}+\mathrm{5tan}\:\frac{\mathrm{4}}{{x}}\right)}{{x}\:\left(\mathrm{1}−\mathrm{cos}\:\frac{\mathrm{6}}{{x}}\right)}\:=? \\ $$ Commented by john_santu last updated on 22/Oct/21 $$\:{let}\:\frac{\mathrm{1}}{{x}}\:=\:{u}\:{and}\:{u}\rightarrow\mathrm{0} \\ $$$$\:\underset{{u}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3}{u}\:\mathrm{ln}\:\left(\mathrm{1}+\mathrm{5tan}\:\mathrm{4}{u}\right)}{\mathrm{1}−\mathrm{cos}\:\mathrm{6}{u}} \\…
Question Number 26223 by abdo imad last updated on 22/Dec/17 $${let}\:{put}\:\xi\left({x}\right)=\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\frac{\mathrm{1}}{{n}^{{x}} }\:\:{with}\:{x}>\mathrm{1} \\ $$$${and}\:\:\delta\left({x}\right)\:\:=\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{{x}} }\:\:\:{find}\:{a}\:{relation} \\ $$$${between}\:\xi\left({x}\right)\:{and}\:\delta\left({x}\right). \\ $$ Commented…
Question Number 26222 by abdo imad last updated on 22/Dec/17 $${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right){n}^{\mathrm{3}} }\:{in}\:{terms}\:{of}\:\xi\left(\mathrm{3}\right) \\ $$$${we}\:{give}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \:\:\frac{\mathrm{1}}{{n}^{{x}} }\:\:{and}\:{x}>\mathrm{1} \\ $$$$\left({zeta}\:{function}\:{of}\:{Rieman}\right) \\ $$ Commented by…
Question Number 26192 by abdo imad last updated on 22/Dec/17 $${find}\:{the}\:{rsdius}\:{of}\:{convergence}\:{for}\:{theserie} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\propto} \:\frac{{x}^{{n}} }{\mathrm{2}{n}+\mathrm{1}}\:{and}\:{calculate}\:{its}\:{sum}\:{s}\left({x}\right) \\ $$$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\:\frac{\mathrm{1}}{\mathrm{2}^{{n}} \left(\mathrm{2}{n}+\mathrm{1}\right)}\:\:. \\ $$ Commented by…
Question Number 26176 by abdo imad last updated on 21/Dec/17 $${find}\:{the}\:{radius}\:{of}\:{convergence}\:{for}\:{the} \\ $$$${serie}\:\sum_{{n}=\mathrm{0}} ^{\propto} {e}^{−\sqrt{{n}}} \:{z}^{{n}} \:\:…{z}\:{from}\:{C}. \\ $$ Commented by abdo imad last updated…
let-put-S-n-k-1-k-n-1-k-k-find-S-n-in-terms-of-H-n-then-lim-n-gt-S-n-H-n-k-1-k-n-1-k-harmonic-serie-
Question Number 26132 by abdo imad last updated on 21/Dec/17 $${let}\:{put}\:{S}_{{n}} \:=\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}} \\ $$$${find}\:{S}_{{n}\:} {in}\:{terms}\:{of}\:\:{H}_{{n}} \:{then}\:{lim}_{{n}−>\propto} \:{S}_{{n}} \\ $$$${H}_{{n}} \:=\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \frac{\mathrm{1}}{{k}}\:\:\:\left(\:{harmonic}\:{serie}\right)…
Question Number 26121 by abdo imad last updated on 20/Dec/17 $${answer}\:{to}\mathrm{26109}\:\:\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:\:\frac{\mathrm{1}}{{k}^{\mathrm{2}} \left({k}+\mathrm{1}\right)^{\mathrm{2}} }\:\:{we}\:{decompose} \\ $$$${F}\left({X}\right)\:\:=\:\:\frac{\mathrm{1}}{{X}^{\mathrm{2}} \left({X}+\mathrm{1}\right)^{\mathrm{2}} }\:=\:\:\frac{{a}}{{X}}\:\:\:+\frac{{b}}{{X}^{\mathrm{2}^{} } }\:\:+\frac{{c}}{{X}+\mathrm{1}}\:\:+\frac{{d}}{\left({X}+\mathrm{1}\right)^{\mathrm{2}} }\:\:{we}\:{find} \\ $$$${F}\left({X}\right)\:\:=\:\:\frac{−\mathrm{2}}{{X}}\:\:+\frac{\mathrm{1}}{{X}^{\mathrm{2}}…
Question Number 26109 by abdo imad last updated on 19/Dec/17 $${let}\:{s}\:{give}\:\:{S}_{{n}} \:=\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:{k}^{−\mathrm{2}\:} .\:\left({k}+\mathrm{1}\right)^{−\mathrm{2}} \\ $$$${find}\:\:{lim}_{{n}−>\propto} \:\:{S}_{{n}} \:\:. \\ $$ Commented by moxhix last…