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Category: Arithmetic

s-i-1-i-s-1-1-2-s-1-3-s-Is-s-gt-0-s-R-1-Can-you-prove-or-prove-otherwise-2-If-s-gt-n-s-R-what-are-the-bounds-of-s-i-e-a-s-b-s-gt-n-

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}…

Question-133764

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}\:…

Question-68212

Question Number 68212 by peter frank last updated on 07/Sep/19 Answered by $@ty@m123 last updated on 07/Sep/19 $${Let}\:{required}\:{equation}\:{of}\:{line}: \\ $$$${y}={m}_{\mathrm{1}} {x}+{c}\:\:\:…..\left(\mathrm{1}\right) \\ $$$${Given}\:{line}:\:\mathrm{4}{x}+\mathrm{3}{y}=\mathrm{21} \\ $$$${Its}\:{slope}:\:{m}_{\mathrm{2}}…

Question-68210

Question Number 68210 by peter frank last updated on 07/Sep/19 Answered by $@ty@m123 last updated on 08/Sep/19 $${Let}\:\frac{\mathrm{sin}\:{A}}{{a}}=\frac{\mathrm{sin}\:{B}}{{b}}=\frac{\mathrm{sin}\:{C}}{{c}}={R} \\ $$$$\Rightarrow\mathrm{sin}\:{A}={aR},\:\mathrm{sin}\:{B}={bR},\:\mathrm{sin}\:{C}={cR}\:…\left(\mathrm{1}\right) \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{sin}\:\left({A}−{B}\right)\mathrm{sin}\:{C}}{\mathrm{1}+\mathrm{cos}\:\left({A}−{B}\right)\mathrm{cos}\:{C}} \\ $$$$=\:\frac{\mathrm{sin}\:\left({A}−{B}\right)\mathrm{sin}\:\left({A}+{B}\right)}{\mathrm{1}−\mathrm{cos}\:\left({A}−{B}\right)\mathrm{cos}\:\left({A}+{B}\right)} \\…

Question-68203

Question Number 68203 by peter frank last updated on 07/Sep/19 Commented by Cmr 237 last updated on 07/Sep/19 $$\eth\mathrm{f}=\frac{\eth\mathrm{f}}{\eth\mathrm{x}}+\frac{\eth\mathrm{f}}{\eth\mathrm{y}}+\frac{\eth\mathrm{f}}{\eth\mathrm{z}} \\ $$$$\:\:\:\:=\mathrm{2xy}+\mathrm{cosh}\left(\mathrm{yz}\right)+\mathrm{x}^{\mathrm{2}} +\mathrm{xzsinh}\left(\mathrm{yz}\right)+\mathrm{xysinh}\left(\mathrm{yz}\right) \\ $$ Commented…

Question-68191

Question Number 68191 by peter frank last updated on 06/Sep/19 Answered by mind is power last updated on 06/Sep/19 $${sin}\left({a}−{b}\right)={sin}\left({a}\right){cos}\left({b}\right)−{cos}\left({a}\right){sin}\left({b}\right) \\ $$$$\frac{{sin}\left({a}−{b}\right)}{{cos}\left({a}\right){cos}\left({b}\right)}={tg}\left({a}\right)−{tg}\left({b}\right) \\ $$$$\Rightarrow\frac{{sin}\left({a}−{b}\right)}{{cos}\left({a}\right){cos}\left({b}\right)}={tg}\left({b}\right)\left({k}−\mathrm{1}\right) \\…