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Category: Algebra

What-is-nth-term-1-2-5-4-15-8-37-16-83-32-

Question Number 3003 by Rasheed Soomro last updated on 02/Dec/15 $${What}\:{is}\:{nth}\:{term}? \\ $$$$\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{5}}{\mathrm{4}},\frac{\mathrm{15}}{\mathrm{8}},\frac{\mathrm{37}}{\mathrm{16}},\frac{\mathrm{83}}{\mathrm{32}}… \\ $$ Commented by Rasheed Soomro last updated on 05/Dec/15 $$\underset{−} {\mathcal{HIGH}{ly}}\:\mathcal{A}{ppriciate}\:\mathcal{Y}{our}\:\overset{\mathcal{VALUEABLE}}…

Calculate-lim-n-k-2-n-k-3-1-k-3-1-

Question Number 134067 by bramlexs22 last updated on 27/Feb/21 $$\:\mathrm{Calculate}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{\mathrm{k}=\mathrm{2}} {\overset{\mathrm{n}} {\prod}}\:\frac{\mathrm{k}^{\mathrm{3}} −\mathrm{1}}{\mathrm{k}^{\mathrm{3}} +\mathrm{1}}\:=\:? \\ $$ Answered by john_santu last updated on 27/Feb/21 $$\:{since}\:\frac{{k}^{\mathrm{3}}…

If-p-gt-1-and-q-gt-1-what-can-be-said-about-the-convergence-of-n-2-1-n-p-ln-n-q-a-always-converges-b-always-diverges-c-may-converges-or-diverges-

Question Number 134060 by benjo_mathlover last updated on 27/Feb/21 $$\mathrm{If}\:\mathrm{p}>\mathrm{1}\:\mathrm{and}\:\mathrm{q}>\mathrm{1}\:\mathrm{what}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{said}\:\mathrm{about}\:\mathrm{the}\:\mathrm{convergence}\: \\ $$$$\mathrm{of}\:\underset{\mathrm{n}=\mathrm{2}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{p}} .\left(\mathrm{ln}\:\mathrm{n}\right)^{\mathrm{q}} }\:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{always}\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{always}\:\mathrm{diverges} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{may}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverges} \\…

If-a-n-is-a-convergent-series-of-nonnegative-terms-what-can-be-said-about-a-n-a-n-1-a-always-converges-b-always-diverges-c-may-converges-or-diverge-

Question Number 134061 by benjo_mathlover last updated on 27/Feb/21 $$\mathrm{If}\:\Sigma\:\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{series}\:\mathrm{of} \\ $$$$\mathrm{nonnegative}\:\mathrm{terms},\mathrm{what}\:\mathrm{can}\:\mathrm{be} \\ $$$$\mathrm{said}\:\mathrm{about}\:\Sigma\:\mathrm{a}_{\mathrm{n}} .\mathrm{a}_{\mathrm{n}+\mathrm{1}} \:? \\ $$$$\left(\mathrm{a}\right)\:\mathrm{always}\:\mathrm{converges} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{always}\:\mathrm{diverges} \\ $$$$\left(\mathrm{c}\right)\:\mathrm{may}\:\mathrm{converges}\:\mathrm{or}\:\mathrm{diverge} \\ $$…

Question-134052

Question Number 134052 by shaker last updated on 27/Feb/21 Answered by mathmax by abdo last updated on 27/Feb/21 $$\mathrm{I}\:=\int\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{6}} \:+\mathrm{3}}\mathrm{dx}\:\:\Rightarrow\mathrm{I}\:=\int\:\:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{x}^{\mathrm{6}} \:+\left(\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{6}}} \right)^{\mathrm{6}} }\mathrm{dx}\:=_{\mathrm{x}=\mathrm{3}^{\frac{\mathrm{1}}{\mathrm{6}}}…

Prove-that-d-dx-e-x-e-x-Assume-that-you-do-not-know-that-the-above-statement-is-true-

Question Number 2980 by Filup last updated on 02/Dec/15 $$\mathrm{Prove}\:\mathrm{that}\:\frac{{d}}{{dx}}\left({e}^{{x}} \right)={e}^{{x}} \\ $$$$\mathrm{Assume}\:\mathrm{that}\:\mathrm{you}\:\mathrm{do}\:\mathrm{not}\:\mathrm{know}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{above}\:\mathrm{statement}\:\mathrm{is}\:\mathrm{true}. \\ $$ Answered by RasheedAhmad last updated on 02/Dec/15 $${e}^{{x}}…

how-many-zeros-has-the-number-1000-at-the-end-and-what-is-the-last-digit-before-these-zeros-

Question Number 134034 by mr W last updated on 27/Feb/21 $${how}\:{many}\:{zeros}\:{has}\:{the}\:{number} \\ $$$$\mathrm{1000}!\:{at}\:{the}\:{end}?\:{and}\:{what}\:{is}\:{the} \\ $$$${last}\:{digit}\:{before}\:{these}\:{zeros}? \\ $$ Answered by floor(10²Eta[1]) last updated on 27/Feb/21 $$\lfloor\frac{\mathrm{1000}}{\mathrm{5}}\rfloor+\lfloor\frac{\mathrm{1000}}{\mathrm{5}^{\mathrm{2}}…

solve-x-3-2-x-5-

Question Number 134005 by mr W last updated on 26/Feb/21 $${solve}\:{x}^{\mathrm{3}} −\mathrm{2}\lfloor{x}\rfloor=\mathrm{5} \\ $$ Answered by MJS_new last updated on 26/Feb/21 $${x}={i}\left[\mathrm{nteger}\:\mathrm{part}\right]+{f}\left[\mathrm{ractal}\:\mathrm{part}\right] \\ $$$$\left({i}+{f}\right)^{\mathrm{3}} −\mathrm{2}{i}=\mathrm{5}…