Question Number 2702 by Filup last updated on 25/Nov/15 $$\zeta\left({s}\right)=\underset{{i}=\mathrm{1}} {\overset{\infty} {\sum}}{i}^{−{s}} =\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{{s}} }+\frac{\mathrm{1}}{\mathrm{3}^{{s}} }+… \\ $$$$ \\ $$$$\mathrm{Is}\:\zeta\left({s}\right)>\mathrm{0}\forall{s}\in\mathbb{R}? \\ $$$$\mathrm{1}.\:\mathrm{Can}\:\mathrm{you}\:\mathrm{prove},\:\mathrm{or}\:\mathrm{prove}\:\mathrm{otherwise}? \\ $$$$\mathrm{2}.\:\mathrm{If}\:\zeta\left({s}\right)>{n},\:{s}\in\mathbb{R},\:\mathrm{what}\:\mathrm{are}\:\mathrm{the}\:\mathrm{bounds} \\ $$$$\mathrm{of}\:{s}?\:\mathrm{i}.\mathrm{e}.\:\:{a}\leqslant{s}\leqslant{b}\::\:\zeta\left({s}\right)>{n}…
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Question Number 68237 by smartsmith459@gmail.com last updated on 07/Sep/19 $$\mathrm{1}.\:{find}\:{the}\:{equation}\:{of}\:{the}\:{line}\:{making}\:{an}\:{angle}\:{of}\:\mathrm{135}^{°\:} {with}\:{O}_{{x}\:} \:{and}\:{passing}\:{through}\:{thepoints}?\left(−\mathrm{2},\mathrm{5}\right) \\ $$$$\mathrm{2}.\:{find}\:{the}\:{slope}\:{of}\:{the}\:{line}\:{through}\:{the}\:{points}\:\left(\mathrm{5},\mathrm{3}\right){and}\:\left(\mathrm{7},\mathrm{2}\right).\:{find}\:\left({i}\right)\:{the}\:{perpendicular}\:{form}\:\left({ii}\right)\:{find}\:{the}\:{intercept}\:{form}\:{of}\:{its}\:{equation}. \\ $$$$\mathrm{3}.\:{Determine}\:{the}\:{gradient}\:{of}\:{the}\:{straight}\:{line}\:{graph}\:{passing}\:{through}\:{the}\:{co}-{ordinates}: \\ $$$$\left({i}\right)\:\left(\mathrm{2},\mathrm{7}\right)\:{and}\:\left(−\mathrm{3},\mathrm{4}\right) \\ $$$$\left({ii}\right)\:\left(\frac{\mathrm{1}\:}{\mathrm{4}},\:\frac{-\mathrm{3}}{\mathrm{4}}\right)\:{and}\:\left(\frac{-\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{5}}{\mathrm{8}}\right). \\ $$ Commented by kaivan.ahmadi…
Question Number 68234 by Mikael last updated on 07/Sep/19 Commented by kaivan.ahmadi last updated on 07/Sep/19 $$\mathrm{3}^{{x}+\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{3}^{{x}−\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{4}^{{x}} +\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} \Rightarrow \\ $$$$\mathrm{3}^{{x}−\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{3}+\mathrm{1}\right)=\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} \left(\mathrm{2}+\mathrm{1}\right)\Rightarrow…
Question Number 2699 by abcd last updated on 25/Nov/15 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when} \\ $$$$\mathrm{3}^{\mathrm{215}} \:\mathrm{is}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{43}. \\ $$ Answered by RasheedAhmad last updated on 25/Nov/15 $$ \\ $$$${Since}\:\left(\mathrm{3},\mathrm{43}\right)=\mathrm{1}\:{and}\:\mathrm{43}\:{is}\:{prime}…
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Question Number 133771 by bramlexs22 last updated on 24/Feb/21 $$\mathrm{A}\:\:\mathrm{circle}\:\mathrm{with}\:\mathrm{radius}\:\mathrm{1}\:\mathrm{centered} \\ $$$$\mathrm{at}\:\left(\mathrm{0},\mathrm{0}\right)\:\mathrm{is}\:\mathrm{conjoined}\:\mathrm{to}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\mathrm{of}\:\mathrm{radius}\:\mathrm{2}\:\mathrm{centered}\:\mathrm{at}\:\left(\mathrm{2},\mathrm{0}\right).\: \\ $$$$\mathrm{Forming}\:\mathrm{a}\:\mathrm{single}\:\mathrm{shape}\:.\mathrm{what}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{shape}'\mathrm{s}\:\mathrm{area}? \\ $$ Commented by bramlexs22 last updated…
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Question Number 133765 by Salman_Abir last updated on 24/Feb/21 Answered by JDamian last updated on 24/Feb/21 $${R}_{{e}} =\infty \\ $$ Terms of Service Privacy Policy…
Question Number 133764 by liberty last updated on 24/Feb/21 Answered by EDWIN88 last updated on 24/Feb/21 $$\:\mathrm{let}\:\mathrm{x}\:\mathrm{be}\:\mathrm{a}\:\mathrm{page}\:\mathrm{number}\:\mathrm{was}\:\mathrm{counted}\:\mathrm{twice} \\ $$$$\mathrm{Assuming}\:\mathrm{the}\:\mathrm{pages}\:\mathrm{start}\:\mathrm{counting}\:\mathrm{at}\:\mathrm{1}\:\mathrm{and} \\ $$$$\mathrm{count}\:\mathrm{continously}\:\mathrm{up}\:\mathrm{we}\:\mathrm{use}\:\mathrm{formula} \\ $$$$\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\mathrm{n}\right)+\mathrm{x}\:=\:\mathrm{1999}\: \\ $$$$\Rightarrow\mathrm{x}\:+\frac{\mathrm{n}\left(\mathrm{n}+\mathrm{1}\right)}{\mathrm{2}}\:=\:\mathrm{1999}\:;\:\mathrm{since}\:\mathrm{x}\:\mathrm{is}\:\mathrm{positive}\:…