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Author: Tinku Tara

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Question Number 5158 by 1771727373 last updated on 24/Apr/16 $${why} \\ $$$$\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{1}}{\mathrm{8}}\:+\:………..\:=\:\mathrm{2} \\ $$$$ \\ $$ Answered by FilupSmith last updated on 24/Apr/16 $$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}}+…={S} \\…

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Question Number 5157 by 1771727373 last updated on 24/Apr/16 $$\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+……..={S} \\ $$$$\mathrm{2}+\mathrm{4}+\mathrm{6}+\mathrm{8}+\mathrm{10}+……..=\mathrm{2}{S} \\ $$$${S}\supset\mathrm{2}{S} \\ $$$$\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}+…………+\left(\mathrm{2}{S}\right)={S} \\ $$$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left\{\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}\right\}=+\infty \\ $$$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left\{{n}^{\mathrm{2}} \right\}=+\infty \\ $$$$\left(+\infty\right)+\left(+\infty\right)={S}=\infty…

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Question Number 5153 by 1771727373 last updated on 23/Apr/16 $${proof}\:\:\:\:{e}^{{i}\Theta} ={cos}\left(\Theta\right)+{isin}\left(\Theta\right) \\ $$ Answered by 123456 last updated on 23/Apr/16 $$\mathrm{lets}\:{f}\left(\theta\right)=\mathrm{cos}\:\theta+{i}\mathrm{sin}\:\theta,{f}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$$\frac{\partial{f}}{\partial\theta}=−\mathrm{sin}\:\theta+{i}\mathrm{cos}\:\theta={i}\left(\mathrm{cos}\:\theta+{i}\mathrm{sin}\:\theta\right)={if} \\ $$$$\frac{{df}}{{f}}={id}\theta…

If-we-had-a-maze-which-was-contained-within-the-shape-of-a-square-and-it-has-one-enterence-on-its-edge-and-a-centre-point-from-which-you-start-If-you-move-randomly-you-will-naturally-escape-Now

Question Number 5150 by FilupSmith last updated on 21/Apr/16 $$\mathrm{If}\:\mathrm{we}\:\mathrm{had}\:\mathrm{a}\:\mathrm{maze},\:\mathrm{which}\:\mathrm{was}\:\mathrm{contained} \\ $$$$\mathrm{within}\:\mathrm{the}\:\mathrm{shape}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square},\:\mathrm{and}\:\mathrm{it}\:\mathrm{has} \\ $$$$\mathrm{one}\:\mathrm{enterence}\:\mathrm{on}\:\mathrm{its}\:\mathrm{edge},\:\mathrm{and}\:\mathrm{a}\:\mathrm{centre} \\ $$$$\mathrm{point}\:\mathrm{from}\:\mathrm{which}\:\mathrm{you}\:\mathrm{start},\:\mathrm{If}\:\mathrm{you}\:\mathrm{move} \\ $$$$\mathrm{randomly},\:\mathrm{you}\:\mathrm{will}\:\mathrm{naturally}\:\mathrm{escape}. \\ $$$$ \\ $$$$\mathrm{Now},\:\mathrm{lets}\:\mathrm{say}\:\mathrm{the}\:\mathrm{maze}\:\mathrm{itself}\:\mathrm{is}\:\mathrm{generated} \\ $$$$\mathrm{at}\:\mathrm{completely}\:\mathrm{random}\:\mathrm{odds}. \\…

Let-n-j-q-Z-1-Are-there-triples-n-j-q-such-that-the-following-conditions-are-satisfied-altogether-i-n-j-q-ii-n-2-j-2-q-2-Suppose-then-that-condition-ii-

Question Number 5144 by Yozzii last updated on 19/Apr/16 $${Let}\:{n},{j},{q}\in\left(\mathbb{Z}^{+} −\left\{\mathrm{1}\right\}\right).\:{Are}\:{there}\: \\ $$$${triples}\:\left({n},{j},{q}\right)\:{such}\:{that}\:{the}\:{following} \\ $$$${conditions}\:{are}\:{satisfied}\:{altogether}? \\ $$$$\left({i}\right)\:{n}={j}^{{q}} \:\:\:\: \\ $$$$\left({ii}\right){n}^{\mathrm{2}} ={j}^{\mathrm{2}} +{q}^{\mathrm{2}} \\ $$$$−−−−−−−−−−−−−−−−−−−−−− \\…