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…
Question Number 157123 by cortano last updated on 20/Oct/21 $$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{log}\:_{\mathrm{sin}\:{x}} \left(\mathrm{cos}\:{x}\right)}{\mathrm{log}\:_{\mathrm{sin}\:\left(\frac{{x}}{\mathrm{2}}\right)} \left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right)}=? \\ $$ Commented by john_santu last updated on 21/Oct/21 $${L}=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)\:\mathrm{ln}\:\left(\mathrm{sin}\:\frac{{x}}{\mathrm{2}}\right)}{\mathrm{ln}\:\left(\mathrm{cos}\:\frac{{x}}{\mathrm{2}}\right).\mathrm{ln}\:\left(\mathrm{sin}\:{x}\right)} \\…
Question Number 26020 by abdo imad last updated on 17/Dec/17 $${answer}\:{to}\:\mathrm{25962}…{we}\:{S}=\sum_{{n}=\mathrm{1}} ^{\propto} \:\mathrm{1}/_{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)} \:{and}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{k}={n}} \:\mathrm{1}/_{{k}^{\mathrm{2}} \left({k}+\mathrm{1}\right)} \\ $$$${we}\:{have}\:{S}=\:{lim}_{{n}−>\propto} \:{S}_{{n}} \:{we}\:{decompose}\:{the}\:{the}\:{rational}\:{fraction} \\ $$$${F}\left({X}\right)=\:\:\:\mathrm{1}/_{{X}}…
Question Number 157055 by cortano last updated on 19/Oct/21 Commented by MathSh last updated on 20/Oct/21 $$=\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\boldsymbol{\pi}} {\mathrm{lim}}\frac{\pi^{\boldsymbol{\mathrm{x}}^{\boldsymbol{\pi}} } \centerdot\pi\mathrm{x}^{\boldsymbol{\mathrm{x}}-\mathrm{1}} \centerdot\mathrm{ln}\left(\pi\right)-\mathrm{x}^{\boldsymbol{\pi}^{\boldsymbol{\mathrm{x}}} } \left(\pi^{\boldsymbol{\mathrm{x}}} \centerdot\mathrm{ln}\left(\pi\right)\centerdot\mathrm{ln}\left(\mathrm{x}\right)+\pi^{\boldsymbol{\mathrm{x}}} \centerdot\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{1}}…
Question Number 25962 by abdo imad last updated on 16/Dec/17 $${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{{n}=\propto} \:\:\:\mathrm{1}/_{{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)} \:\:{we}\:{give}\:\:\:\sum_{{n}=\mathrm{1}} ^{{n}=\propto} \mathrm{1}/_{{n}} \mathrm{2}=\:\pi^{\mathrm{2}} /\mathrm{6} \\ $$$${and}\:\:{H}_{{n}} =\mathrm{1}+\mathrm{2}^{−\mathrm{1}} +\mathrm{3}^{−\mathrm{1}} +…+{n}^{−\mathrm{1}} =\:{ln}\left({n}\right)\:+\:{s}\:+\:\theta\left(\mathrm{1}/{n}\right)\:…
Question Number 156993 by amin96 last updated on 18/Oct/21 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{n}} {\underbrace{\left({sin}\left({sin}\left({sin}\ldots\left({sin}\left({x}\right)\right)\ldots\right)}}\:\sqrt{{n}}=?\right.\right. \\ $$$$\mathrm{0}<{x}<\pi \\ $$ Commented by MathSh last updated on 18/Oct/21 $$\mathrm{0}<{x}<\pi\:\:\mathrm{or}\:\:\mathrm{0}\leqslant{x}\leqslant\pi.? \\…