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Question-165349

Question Number 165349 by mkam last updated on 31/Jan/22 Answered by puissant last updated on 31/Jan/22 $$\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} }\:=\:\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)−\left(\mathrm{2}{n}−\mathrm{1}\right)}{\mathrm{1}+\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{2}{n}−\mathrm{1}\right)} \\ $$$$\Rightarrow\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{arctan}\left(\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} }\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{{arctan}\left(\mathrm{2}{n}+\mathrm{1}\right)−{arctan}\left(\mathrm{2}{n}−\mathrm{1}\right)\right\}…

Question-165301

Question Number 165301 by mathdave last updated on 28/Jan/22 Answered by som(math1967) last updated on 29/Jan/22 $$\mathrm{1}.\boldsymbol{{No}}\:\boldsymbol{{of}}\:\boldsymbol{{ways}}\:'\boldsymbol{{median}}'\:\boldsymbol{{be}}\:\boldsymbol{{arranged}} \\ $$$$\:\mathrm{6}!=\mathrm{720} \\ $$$$\mathrm{2}.\:\boldsymbol{{M}}\:\boldsymbol{{begin}} \\ $$$$\:\boldsymbol{{M}}\:\Box\Box\Box\Box\Box\:^{\mathrm{5}} \boldsymbol{{P}}_{\mathrm{5}} =\mathrm{120}…

find-the-sum-n-1-1-2-n-i-1-3-n-

Question Number 165262 by mkam last updated on 28/Jan/22 $$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{sum}}\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\boldsymbol{{n}}} \:+\:\boldsymbol{{i}}\:\left(\:\frac{\mathrm{1}}{\mathrm{3}}\:\right)^{\boldsymbol{{n}}} \: \\ $$ Answered by Mathspace last updated on 28/Jan/22 $$\left.\sum_{{n}=\mathrm{1}} ^{\infty}…

soit-la-serie-de-fonction-n-2-x-n-nx-ln-n-etudie-la-convergence-simple-sur-0-1-

Question Number 165240 by SANOGO last updated on 28/Jan/22 $$\:{soit}\:{la}\:{serie}\:{de}\:{fonction}\:\underset{{n}=\mathrm{2}\:\:\:} {\sum}\frac{{x}^{{n}} }{{nx}+{ln}\left({n}\right)} \\ $$$${etudie}\:{la}\:{convergence}\:{simple}\:{sur}\:\left[\mathrm{0},\mathrm{1}\left[\right.\right. \\ $$ Answered by mindispower last updated on 28/Jan/22 $$\forall{a}\in\left[\mathrm{0},\mathrm{1}\right] \\…

lim-x-1-y-2-x-2-y-2-

Question Number 165208 by Jamshidbek last updated on 27/Jan/22 $$\underset{{x}\rightarrow\underset{\mathrm{y}\rightarrow\mathrm{2}} {\mathrm{1}}} {\mathrm{lim}}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =? \\ $$ Answered by alephzero last updated on 27/Jan/22 $$\underset{\underset{{y}\rightarrow\mathrm{2}} {{x}\rightarrow\mathrm{1}}}…