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Question Number 165402 Answers: 0 Comments: 1

Question Number 165532 Answers: 1 Comments: 0

Question Number 165392 Answers: 2 Comments: 0

lim_(x→+∞) ((e^(1/x) −cos (1/x))/(1−(√(1−(1/x^2 ))))) lim_(x→a) ((x^x −a^a )/(x−a))

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\frac{{e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{cos}\:\frac{\mathrm{1}}{{x}}}{\mathrm{1}−\sqrt{\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }}} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{{x}} −{a}^{{a}} }{{x}−{a}} \\ $$

Question Number 165389 Answers: 2 Comments: 1

Question Number 165385 Answers: 1 Comments: 1

Number of ways..n differrnt things be distributed in r identical boxes so as 1)empty box is allowed 2)empty box not allowed Number of ways...n identical things be distributed to r identical boxes so as 1)empty box allowed 2)empty box not allowed

$${Number}\:{of}\:\:{ways}..{n}\: \\ $$$${differrnt}\:\:{things}\:{be}\:{distributed} \\ $$$${in}\:{r}\:{identical}\:{boxes}\:{so}\:{as} \\ $$$$\left.\mathrm{1}\right){empty}\:{box}\:{is}\:{allowed} \\ $$$$\left.\mathrm{2}\right){empty}\:{box}\:{not}\:{allowed} \\ $$$${Number}\:{of}\:{ways}...{n}\:{identical} \\ $$$${things}\:{be}\:{distributed}\:{to}\:{r} \\ $$$${identical}\:{boxes}\:{so}\:{as} \\ $$$$\left.\mathrm{1}\right){empty}\:{box}\:{allowed} \\ $$$$\left.\mathrm{2}\right){empty}\:{box}\:{not}\:{allowed} \\ $$$$ \\ $$

Question Number 165383 Answers: 0 Comments: 0

Find the value of this expression (−1)(2^2 −1)+(((3^2 −2))/(2!)) − (((4^2 −3))/(3!)) + …+ (((2019^2 −2018))/(2018!))

$${Find}\:\:{the}\:\:{value}\:\:{of}\:\:{this}\:\:{expression} \\ $$$$\left(−\mathrm{1}\right)\left(\mathrm{2}^{\mathrm{2}} −\mathrm{1}\right)+\frac{\left(\mathrm{3}^{\mathrm{2}} −\mathrm{2}\right)}{\mathrm{2}!}\:−\:\frac{\left(\mathrm{4}^{\mathrm{2}} −\mathrm{3}\right)}{\mathrm{3}!}\:+\:\ldots+\:\frac{\left(\mathrm{2019}^{\mathrm{2}} −\mathrm{2018}\right)}{\mathrm{2018}!} \\ $$

Question Number 165382 Answers: 0 Comments: 1

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Question Number 165380 Answers: 1 Comments: 0

∫_0 ^( (π/2)) (( x^( 3) )/(sin^( 2) (x)))dx=^? (3/8) (π^( 2) ln(4)−7ζ(3))

$$ \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\:{x}^{\:\mathrm{3}} }{{sin}^{\:\mathrm{2}} \left({x}\right)}{dx}\overset{?} {=}\:\frac{\mathrm{3}}{\mathrm{8}}\:\left(\pi^{\:\mathrm{2}} {ln}\left(\mathrm{4}\right)−\mathrm{7}\zeta\left(\mathrm{3}\right)\right) \\ $$

Question Number 165376 Answers: 1 Comments: 0

Verify wether f is invertible f (x) = (1+2x)^3

$$\mathrm{Verify}\:\mathrm{wether}\:{f}\:\mathrm{is}\:\mathrm{invertible}\: \\ $$$${f}\:\left({x}\right)\:=\:\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{3}} \\ $$

Question Number 165375 Answers: 1 Comments: 0

Question Number 165372 Answers: 1 Comments: 0

lim_(x→+∞) ln (1+2^x )ln (1+(3/x))

$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}ln}\:\left(\mathrm{1}+\mathrm{2}^{{x}} \right)\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{3}}{{x}}\right) \\ $$

Question Number 165370 Answers: 1 Comments: 1

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Question Number 165361 Answers: 1 Comments: 2

Obtain a general formula for the sequence (2/3),(4/5),(8/9),((16)/(17)),((32)/(33)),... assuming the sequence continues in that pattern.

$$\mathrm{Obtain}\:\mathrm{a}\:\mathrm{general}\:\mathrm{formula}\:\mathrm{for} \\ $$$$\mathrm{the}\:\mathrm{sequence} \\ $$$$\:\frac{\mathrm{2}}{\mathrm{3}},\frac{\mathrm{4}}{\mathrm{5}},\frac{\mathrm{8}}{\mathrm{9}},\frac{\mathrm{16}}{\mathrm{17}},\frac{\mathrm{32}}{\mathrm{33}},... \\ $$$$\mathrm{assuming}\:\mathrm{the}\:\mathrm{sequence}\:\mathrm{continues}\:\mathrm{in}\:\mathrm{that} \\ $$$$\mathrm{pattern}. \\ $$

Question Number 165357 Answers: 1 Comments: 2

Question Number 165355 Answers: 1 Comments: 0

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Question Number 165339 Answers: 1 Comments: 0

# Advanced Calculus # Let , f : R → Q is a continuous function . prove that ” f ” is a constant function . ■ m.n ∗ Adopted from mathematical analysis book ∗ −−−−−−−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:#\:{Advanced}\:\:\:{Calculus}\:#\:\:\: \\ $$$$\:\:\:\:\:\mathrm{L}{et}\:,\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{Q}\:\:{is}\:\:{a}\:\:{continuous} \\ $$$$\:\:\:\:\:\:{function}\:\:.\:{prove}\:{that}\:\:''\:{f}\:''\:{is}\:{a} \\ $$$$\:\:\:\:\:\:\:{constant}\:{function}\:.\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$$$\:\:\:\:\:\:\:\ast\:{Adopted}\:{from}\:{mathematical}\:{analysis}\:{book}\:\:\ast\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−−−−−−− \\ $$$$ \\ $$

Question Number 165329 Answers: 2 Comments: 0

prove ( n∈ N ) 3(n+1) ∣ n^( 3) + (n+1)^( 3) + (n+2 )^( 3)

$$ \\ $$$$\:\:\:\:{prove}\:\:\:\:\:\:\:\:\:\:\left(\:{n}\in\:\mathbb{N}\:\right) \\ $$$$\:\:\:\:\mathrm{3}\left({n}+\mathrm{1}\right)\:\mid\:{n}^{\:\mathrm{3}} \:+\:\left({n}+\mathrm{1}\right)^{\:\mathrm{3}} +\:\left({n}+\mathrm{2}\:\right)^{\:\mathrm{3}} \\ $$$$ \\ $$

Question Number 165328 Answers: 3 Comments: 0

prove that Nice Integral 𝛗=∫_0 ^( 1) (( tan^( −1) (x^( (3/2)) ))/x^( 2) ) dx =((π + (√3) ln(7 +4(√3) ))/4) ■ m.n −−−−−−−−−

$$ \\ $$$$\:\:\:\:\:{prove}\:{that} \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\mathscr{N}{ice}\:\:\:\mathscr{I}{ntegral} \\ $$$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tan}^{\:−\mathrm{1}} \:\left({x}^{\:\frac{\mathrm{3}}{\mathrm{2}}} \right)}{{x}^{\:\mathrm{2}} }\:{dx}\:\:=\frac{\pi\:+\:\sqrt{\mathrm{3}}\:{ln}\left(\mathrm{7}\:+\mathrm{4}\sqrt{\mathrm{3}}\:\right)}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n} \\ $$$$\:\:\:\:\:\:−−−−−−−−−\:\:\: \\ $$

Question Number 165322 Answers: 1 Comments: 2

who can prove that 2^n −1produces a prime number when n is a prime number

$${who}\:{can}\:{prove}\:{that}\:\mathrm{2}^{{n}} −\mathrm{1}{produces} \\ $$$${a}\:{prime}\:{number}\:{when}\:\:{n}\:\:{is}\:{a}\: \\ $$$${prime}\:{number} \\ $$

Question Number 165321 Answers: 2 Comments: 0

Question Number 165320 Answers: 1 Comments: 0

Ω = ∫_0 ^( (π/4)) cos (2x ).e^( ⌊ sin(x)+ cos(x) ⌋) dx ⌊ x ⌋= max { m ∈ Z ∣ m ≤ x } −−−−

$$ \\ $$$$\Omega\:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:{cos}\:\left(\mathrm{2}{x}\:\right).{e}^{\:\lfloor\:{sin}\left({x}\right)+\:{cos}\left({x}\right)\:\rfloor} {dx} \\ $$$$\:\:\:\lfloor\:{x}\:\rfloor=\:{max}\:\left\{\:{m}\:\in\:\mathbb{Z}\:\mid\:\:{m}\:\leqslant\:{x}\:\right\} \\ $$$$\:\:\:\:\:\:\:\:−−−− \\ $$

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