Category: Number Theory

Question Number 10671 by Saham last updated on 22/Feb/17 $$\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}5{\left(\frac{1}{4}\right)}^{\mathrm{n}-1}$$ Answered by FilupS last updated on 22/Feb/17 $${S}=\mathrm{5}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{4}^{{n}−\mathrm{1}} }…

Question Number 10665 by FilupS last updated on 22/Feb/17 $$n\mathrm{th}\mathrm{prime}=p_n$$$$n\mathrm{th}\mathrm{non}-\mathrm{prime}=q_n$$$$$$$$\mathrm{Determine}\mathrm{if}{q}_n>p_n\mathrm{for}\mathrm{\forall }n⩾2$$ Commented by FilupS last updated…

Question Number 10662 by FilupS last updated on 22/Feb/17 $$\mathrm{determine}\mathrm{if}:$$$$\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{5^s}+\dots ⩾\frac{1}{1^s}+\frac{1}{4^s}+\frac{1}{6^s}+\dots$$$$\mathrm{or}:$$$$\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\in\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}}…

Question Number 141694 by Dwaipayan Shikari last updated on 22/May/21 $$log\left(\frac{\sqrt{5}+1}{10}9e^{\gamma }\right)=\frac{\zeta \left(2\right)}{2}\left(\frac{1^2+9^2}{{10}^2}\right)-\frac{\zeta \left(3\right)}{3}\left(\frac{1^3+9^3}{{10}^3}\right)+\frac{\zeta \left(4\right)}{4}\left(\frac{1^4+9^4}{{10}^4}\right)-\dots$$$$\gamma =EulerMascheroniConstant$$…

Question Number 10539 by malwaan last updated on 17/Feb/17 $$provethat$$$$\sqrt{2+\stackrel{3}{}\sqrt{3+\dots +\stackrel{1993}{}\sqrt{1993}}}<2$$ Answered by mrW1 last updated on 17/Feb/17 $${function}\:{y}=\left({x}+{a}\right)^{\frac{\mathrm{1}}{{x}}} \:{with}\:{a}>\mathrm{1}\:{is}…