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Let-say-r-n-k-0-n-1-r-k-and-r-0-1-With-n-N-and-r-R-1-Show-that-n-1-r-n-1-n-r-n-2-If-m-n-show-that-r-n-r-m-r-m-n-m-3-Espress-r-

Question Number 115534 by Hassen_Timol last updated on 26/Sep/20 $$\mathrm{Let}\:\mathrm{say}\:{r}^{\left({n}\right)} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\left({r}−{k}\right)\:\mathrm{and}\:{r}^{\left(\mathrm{0}\right)} =\mathrm{1} \\ $$$$\mathrm{With}\:{n}\in\mathbb{N}\:\mathrm{and}\:{r}\in\mathbb{R}… \\ $$$$\mathrm{1}.\:\:\:\mathrm{Show}\:\mathrm{that}\:\left({n}−\mathrm{1}−{r}\right)^{\left({n}\right)} \:=\:\left(−\mathrm{1}\right)^{\left({n}\right)} \left({r}\right)^{\left({n}\right)} \\ $$$$\mathrm{2}.\:\mathrm{If}\:{m}\leqslant{n},\:\mathrm{show}\:\mathrm{that}\:\:\frac{{r}^{\left({n}\right)} }{{r}^{\left({m}\right)} }=\left({r}−{m}\right)^{\left({n}−{m}\right)} \\…

sec-x-dx-x-x-dx-lim-n-1-1-n-2-1-n-2-4-n-2-1-n-3-9-n-2-1-n-n-2n-2-1-n-n-

Question Number 115498 by bobhans last updated on 26/Sep/20 $$\int\:\mathrm{sec}\:{x}\:{dx}\:? \\ $$$$\int\:\sqrt{{x}−\sqrt{{x}}}\:{dx}\:? \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\sqrt[{{n}\:}]{\mathrm{1}+\sqrt{\mathrm{1}+{n}^{\mathrm{2}} }}\right)\left(\sqrt[{{n}\:}]{\mathrm{2}+\sqrt{\mathrm{4}+{n}^{\mathrm{2}} }}\right)\left(\sqrt[{{n}\:}]{\mathrm{3}+\sqrt{\mathrm{9}+{n}^{\mathrm{2}} }}\right)…\left(\sqrt[{{n}\:}]{{n}+\sqrt{\mathrm{2}{n}^{\mathrm{2}} }}\right)}{{n}}? \\ $$ Commented by bobhans last…

Question-115472

Question Number 115472 by Khalmohmmad last updated on 26/Sep/20 Commented by malwaan last updated on 26/Sep/20 $${x}\rightarrow−\infty \\ $$$$\mid{x}−\mathrm{2}\mid=−{x}+\mathrm{2} \\ $$$$\mid{x}+\mathrm{1}\mid=−{x}−\mathrm{1} \\ $$$$\therefore\:\underset{{x}\rightarrow−\infty} {{lim}}\:\frac{\mathrm{3}{x}−{x}+\mathrm{2}}{\mathrm{5}{x}+{x}+\mathrm{1}}\: \\…

For-angles-a-b-c-R-with-a-b-c-pi-prove-the-following-identities-tan-1-a-2-tan-1-b-2-tan-1-c-2-tan-a-2-tan-b-2-tan-c-2-1-Help-

Question Number 115471 by manuel2456 last updated on 26/Sep/20 $${For}\:{angles}\:{a},{b},{c}\in\mathbb{R}\:{with}\:{a}+{b}+{c}=\pi,\: \\ $$$${prove}\:{the}\:{following}\:{identities}: \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{{a}}{\mathrm{2}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{b}}{\mathrm{2}}\right)+\mathrm{tan}^{−\mathrm{1}} \left(\frac{{c}}{\mathrm{2}}\right)=\left(\mathrm{tan}\left(\frac{{a}}{\mathrm{2}}\right)\mathrm{tan}\left(\frac{{b}}{\mathrm{2}}\right)\mathrm{tan}\left(\frac{{c}}{\mathrm{2}}\right)\right)^{−\mathrm{1}} \\ $$$$\mathrm{H}{elp} \\ $$$$ \\ $$ Commented by…

Question-115438

Question Number 115438 by A8;15: last updated on 25/Sep/20 Answered by Olaf last updated on 26/Sep/20 $${the}\:{function}\:{is}\:{odd} \\ $$$${and}\:{the}\:{domain}\:{is}\:{symmetrical} \\ $$$${so}\:{the}\:{result}\:{is}\:\mathrm{0}. \\ $$$$\boldsymbol{\mathrm{Sorry}}\:\boldsymbol{\mathrm{I}}'\boldsymbol{\mathrm{m}}\:\boldsymbol{\mathrm{wrong}}. \\ $$$$\boldsymbol{\mathrm{see}}\:\boldsymbol{\mathrm{mister}}\:\boldsymbol{\mathrm{abdo}}.…