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Category: Arithmetic

a-Find-the-sum-given-by-S-n-1-1-3-1-3-5-1-5-7-1-2n-1-2n-1-b-find-the-limit-of-S-n-as-n-

Question Number 8764 by tawakalitu last updated on 26/Oct/16 $$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{given}\:\mathrm{by} \\ $$$$\mathrm{S}_{\mathrm{n}} \:=\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{5}.\mathrm{7}}\:+\:…\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of}\:\:\:\mathrm{S}_{\mathrm{n}} \:\:\mathrm{as}\:\:\mathrm{n}\:\rightarrow\:\infty \\ $$ Commented by sou1618 last updated on 26/Oct/16…

Question-139814

Question Number 139814 by help last updated on 01/May/21 Answered by physicstutes last updated on 01/May/21 $${PV}\:=\:{nRT} \\ $$$$\mathrm{and}\:{P}\:\mathrm{increases}\:\mathrm{at}\:\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{0}.\mathrm{10}\:\mathrm{atm}/\mathrm{min}\:\mathrm{and}\:{V}\:\mathrm{is}\:\mathrm{decreasing}\:\mathrm{at} \\ $$$$\:\mathrm{a}\:\mathrm{rate}\:\mathrm{of}\:\mathrm{0}.\mathrm{15}\:\mathrm{L}/\mathrm{min}\:\mathrm{when}\:{n}\:=\:\mathrm{10}\: \\ $$$${T}\:=\:\frac{{PV}}{{nR}}\:\Rightarrow\:\frac{{dT}}{{dt}}\:=\:\frac{\mathrm{1}}{{nR}}\left({V}_{\mathrm{0}} \frac{{dP}}{{dt}}+{P}_{\mathrm{0}} \frac{{dV}}{{dt}}\right)…

Problem-15-Find-the-sum-of-S-3-1-2-3-4-2-3-4-5-3-4-5-2016-2014-2015-2016-

Question Number 8554 by Sopheak last updated on 16/Oct/16 $${Problem}\:.\mathrm{15} \\ $$$${Find}\:{the}\:{sum}\:{of} \\ $$$${S}=\:\frac{\mathrm{3}}{\mathrm{1}!+\mathrm{2}!+\mathrm{3}!}+\frac{\mathrm{4}}{\mathrm{2}!+\mathrm{3}!+\mathrm{4}!}+\frac{\mathrm{5}}{\mathrm{3}!+\mathrm{4}!+\mathrm{5}!}+…+\frac{\mathrm{2016}}{\mathrm{2014}!+\mathrm{2015}!+\mathrm{2016}!} \\ $$$$\: \\ $$ Commented by Yozzias last updated on 16/Oct/16…

Question-8558

Question Number 8558 by Sopheak last updated on 16/Oct/16 Commented by FilupSmith last updated on 16/Oct/16 $${S}_{{n}} =\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}^{\mathrm{4}} }{\left(\mathrm{2}{k}−\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)} \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{2}{k}−\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)}=\frac{{A}}{\mathrm{2}{k}−\mathrm{1}}+\frac{{B}}{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\mathrm{1}={A}\left(\mathrm{2}{k}+\mathrm{1}\right)+{B}\left(\mathrm{2}{k}−\mathrm{1}\right)…

Question-8553

Question Number 8553 by Sopheak last updated on 16/Oct/16 Commented by Yozzias last updated on 16/Oct/16 $$\frac{\mathrm{1}}{\mathrm{n}!}−\frac{\mathrm{1}}{\left(\mathrm{n}+\mathrm{1}\right)!}=\frac{\mathrm{1}}{\mathrm{n}!}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{n}!}\left(\frac{\mathrm{n}+\mathrm{1}−\mathrm{1}}{\mathrm{n}+\mathrm{1}}\right)=\frac{\mathrm{n}}{\mathrm{n}!\left(\mathrm{n}+\mathrm{1}\right)}=\frac{\mathrm{n}}{\left(\mathrm{n}+\mathrm{1}\right)!} \\ $$ Answered by Yozzias last updated on…

Question-139548

Question Number 139548 by peter frank last updated on 28/Apr/21 Answered by Dwaipayan Shikari last updated on 28/Apr/21 $$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}} {sin}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{6}} }{dx}=−\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}}…

Prove-this-equation-2-x-1-x-1-2-1-2-2x-1-2-x-1-x-1-

Question Number 73974 by Raxreedoroid last updated on 17/Nov/19 $$\mathrm{Prove}\:\mathrm{this}\:\mathrm{equation} \\ $$$$\frac{\mathrm{2}^{{x}−\mathrm{1}} \left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)!}{\frac{\mathrm{1}}{\mathrm{2}}!}=\frac{\left(\mathrm{2}{x}−\mathrm{1}\right)!}{\mathrm{2}^{{x}−\mathrm{1}} \left({x}−\mathrm{1}\right)!} \\ $$ Answered by mind is power last updated on 17/Nov/19…