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Author: Tinku Tara

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Question Number 69247 by Rio Michael last updated on 21/Sep/19 $${show}\:{that}\: \\ $$$$\:{c}\mid{a}\:\Leftrightarrow\:−{c}\mid{a}. \\ $$ Commented by kaivan.ahmadi last updated on 22/Sep/19 $${c}\mid{a}\Rightarrow{a}={cx};\:\exists{x}\in\mathbb{Z} \\ $$$$\Rightarrow{a}=−{c}\left(−{x}\right)…

lets-say-we-have-a-function-f-x-The-tangent-line-at-x-n-makes-an-angle-with-the-x-axis-What-is-the-rate-of-change-of-the-angle-as-we-change-the-value-of-x-Suppose-f-x-x-2-

Question Number 3707 by Filup last updated on 19/Dec/15 $$\mathrm{lets}\:\mathrm{say}\:\mathrm{we}\:\mathrm{have}\:\mathrm{a}\:\mathrm{function}\:{f}\left({x}\right). \\ $$$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{at}\:{x}={n}\:\mathrm{makes}\:\mathrm{an} \\ $$$$\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:{x}−\mathrm{axis}. \\ $$$$ \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{change}\:\mathrm{of}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{as}\:\mathrm{we}\:\mathrm{change}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}? \\ $$$$ \\ $$$$\mathrm{Suppose}\:{f}\left({x}\right)={x}^{\mathrm{2}} \\…

Let-consider-K-0-1-1-x-a-1-x-b-1-x-c-x-1-lnx-dx-prove-that-e-K-a-b-a-c-b-c-a-b-c-a-b-c-

Question Number 69241 by ~ À ® @ 237 ~ last updated on 21/Sep/19 $$\:{Let}\:{consider}\:{K}=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\left(\mathrm{1}−{x}^{{a}} \right)\left(\mathrm{1}−{x}^{{b}} \right)\left(\mathrm{1}−{x}^{{c}} \right)}{\left({x}−\mathrm{1}\right){lnx}}{dx}\: \\ $$$${prove}\:{that}\: \\ $$$${e}^{{K}} =\:\frac{\left({a}+{b}\right)!\left({a}+{c}\right)!\left({b}+{c}\right)!}{{a}!{b}!{c}!\left({a}+{b}+{c}\right)!}\:\:…

Use-Residus-theorem-to-prove-that-a-gt-0-n-0-1-n-2-a-2-1-2-pi-ash-pia-1-a-2-and-n-0-1-n-n-2-a-2-1-2-pi-a-th-pia-1-a-2-Assume

Question Number 69236 by ~ À ® @ 237 ~ last updated on 21/Sep/19 $${Use}\:\:{Residus}\:{theorem}\:{to}\:{prove}\:{that}\:\forall\:{a}>\mathrm{0}\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{n}^{\mathrm{2}} +{a}^{\mathrm{2}} }\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{\pi}{{ash}\left(\pi{a}\right)}\:\:\:−\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\right) \\ $$$${and}\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{2}}…