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Question Number 184925 Answers: 1 Comments: 1

Question Number 184920 Answers: 1 Comments: 0

Investigate the series (1/(1×2))+(1/(2×3))+(1/(3×4))+(1/(4×5))+... Does it Converges or Diverges?

$$\mathrm{Investigate}\:\mathrm{the}\:\mathrm{series} \\ $$$$\frac{\mathrm{1}}{\mathrm{1}×\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}×\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{3}×\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{4}×\mathrm{5}}+... \\ $$$$\mathrm{Does}\:\mathrm{it}\:\mathrm{Converges}\:\mathrm{or}\:\mathrm{Diverges}? \\ $$

Question Number 184918 Answers: 2 Comments: 2

Show that 1+(1/2)+(1/3)+(1/4)+(1/5)+(1/6)+... is not convergent Hi

$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{6}}+...\:\mathrm{is}\: \\ $$$$\mathrm{not}\:\mathrm{convergent} \\ $$$$ \\ $$$$\mathrm{Hi} \\ $$

Question Number 184915 Answers: 0 Comments: 2

Consider the series below 1+5+25+125+... Investigate whether it is convergent or divergent. Thanks

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{series}\:\mathrm{below} \\ $$$$\mathrm{1}+\mathrm{5}+\mathrm{25}+\mathrm{125}+...\:\mathrm{Investigate} \\ $$$$\mathrm{whether}\:\mathrm{it}\:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\: \\ $$$$\mathrm{divergent}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Thanks} \\ $$

Question Number 184914 Answers: 0 Comments: 1

Lindsey randomly selects 2 distinct days each lveek to go to the gym. Jess randomly selects 3 distinct days each tveek to go to the same gym. What is the probability that Lindsey and Jess will both be at the gym on the same day at least once in any given week? Express your anslver as a common fraction.

$$ \\ $$ Lindsey randomly selects 2 distinct days each lveek to go to the gym. Jess randomly selects 3 distinct days each tveek to go to the same gym. What is the probability that Lindsey and Jess will both be at the gym on the same day at least once in any given week? Express your anslver as a common fraction.

Question Number 184913 Answers: 0 Comments: 0

Question Number 184903 Answers: 1 Comments: 0

lim_(x→0) ((e^x +e^(−x) −2)/x^2 )=?? let x=2t { ((x→0)),((t→0)) :} lim_(x→0) ((e^x +e^(−x) −2)/x^2 )=lim_(t→0) ((e^(2t) +e^(−2t) −2)/(4t^2 ))=(1/4)lim_(t→0) (((e^t −e^(−t) )/t))^2 =(1/4)lim_(t→0) (((e^t −1)/t)+((e^(−t) −1)/(−t)))^2 =(1/4)(1+1)^2 =1

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} −\mathrm{2}}{\mathrm{x}^{\mathrm{2}} }=?? \\ $$$$\mathrm{let}\:\mathrm{x}=\mathrm{2t}\:\begin{cases}{\mathrm{x}\rightarrow\mathrm{0}}\\{\mathrm{t}\rightarrow\mathrm{0}}\end{cases} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{e}^{\mathrm{x}} +\mathrm{e}^{−\mathrm{x}} −\mathrm{2}}{\mathrm{x}^{\mathrm{2}} }=\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{e}^{\mathrm{2t}} +\mathrm{e}^{−\mathrm{2t}} −\mathrm{2}}{\mathrm{4t}^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{4}}\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{e}^{\mathrm{t}} −\mathrm{e}^{−\mathrm{t}} }{\mathrm{t}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}}\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{e}^{\mathrm{t}} −\mathrm{1}}{\mathrm{t}}+\frac{\mathrm{e}^{−\mathrm{t}} −\mathrm{1}}{−\mathrm{t}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{1}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{1} \\ $$

Question Number 184901 Answers: 2 Comments: 1

lim_((2/x)→2) ((100+x)/(101x))

$$\underset{\frac{\mathrm{2}}{{x}}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{100}+{x}}{\mathrm{101}{x}} \\ $$

Question Number 184893 Answers: 1 Comments: 0

find the set of critical points for: 1: f (x) = (x/(⌊ x ⌋+⌊ −x⌋)) −x 2 : g(x) = (x/(⌊ x ⌋ +⌊ −x ⌋)) +x

$$ \\ $$$$\:\:\:\:{find}\:{the}\:{set}\:{of}\: \\ $$$$\:\:\:\:{critical}\:{points}\:{for}: \\ $$$$\:\:\:\mathrm{1}:\:\:\:{f}\:\left({x}\right)\:=\:\frac{{x}}{\lfloor\:{x}\:\rfloor+\lfloor\:−{x}\rfloor}\:−{x}\:\:\:\: \\ $$$$\:\:\:\:\mathrm{2}\::\:\:{g}\left({x}\right)\:=\:\frac{{x}}{\lfloor\:{x}\:\rfloor\:+\lfloor\:−{x}\:\rfloor}\:+{x} \\ $$

Question Number 184891 Answers: 3 Comments: 0

lim_(x→π) ((x−π)/(sinx))=?

$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{{x}−\pi}{{sinx}}=? \\ $$

Question Number 184883 Answers: 1 Comments: 0

lim_(x→2) ((√(x−2))/(x−2))=?

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\frac{\sqrt{{x}−\mathrm{2}}}{{x}−\mathrm{2}}=? \\ $$

Question Number 184882 Answers: 1 Comments: 0

lim_(x→0) ((ln(7x+1))/x)=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{7}{x}+\mathrm{1}\right)}{{x}}=? \\ $$

Question Number 184881 Answers: 1 Comments: 0

lim_(x→0) ((((√2)sinx)/x))^((sin(x/2)cos(x/2))/x) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\sqrt{\mathrm{2}}{sinx}}{{x}}\right)^{\frac{{sin}\frac{{x}}{\mathrm{2}}{cos}\frac{{x}}{\mathrm{2}}}{{x}}} =? \\ $$

Question Number 184880 Answers: 1 Comments: 0

lim_(x→(1/π)) ((π^(−1) −x)/((1−(xπ)^2 )/π^2 ))=?

$$\underset{{x}\rightarrow\frac{\mathrm{1}}{\pi}} {\mathrm{lim}}\:\:\frac{\pi^{−\mathrm{1}} −{x}}{\frac{\mathrm{1}−\left({x}\pi\right)^{\mathrm{2}} }{\pi^{\mathrm{2}} }}=? \\ $$

Question Number 184879 Answers: 1 Comments: 0

lim_(x→(1/4)) 3x=(3/4) x=?

$$\underset{{x}\rightarrow\frac{\mathrm{1}}{\mathrm{4}}} {\mathrm{lim}}\:\mathrm{3}{x}=\frac{\mathrm{3}}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:{x}=? \\ $$

Question Number 184878 Answers: 1 Comments: 0

lim_(x→0) ((ln(1+sinx))/(sinx))=? with out H′L Roule

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{ln}\left(\mathrm{1}+{sinx}\right)}{{sinx}}=? \\ $$$${with}\:{out}\:{H}'{L}\:{Roule} \\ $$

Question Number 184877 Answers: 1 Comments: 0

Find the sum of the last ten digits: 5^(23) ∙ 2^(2022) − 2023

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{last}\:\mathrm{ten}\:\mathrm{digits}: \\ $$$$\mathrm{5}^{\mathrm{23}} \:\centerdot\:\mathrm{2}^{\mathrm{2022}} \:−\:\mathrm{2023} \\ $$

Question Number 184875 Answers: 1 Comments: 0

Question Number 184873 Answers: 1 Comments: 0

Find the real number satisfying x=(√(1+(√(1+(√(1+x))))))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{real}\:\mathrm{number}\:\mathrm{satisfying} \\ $$$$\:\mathrm{x}=\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{x}}}} \\ $$

Question Number 184872 Answers: 0 Comments: 0

An odd function f(x) whose domain is R satisfies f(x)=f(x+2). When x ∈ (0, 1), f(x)=−2x^2 +ax−2. If f has 2023 zeros in [0, 1011]. Then the range of a can be ? A. [−6, −2(√2)] B. [−4, −2(√2)] C. [−8, −6] D. [−6, −4]

$$\mathrm{An}\:\mathrm{odd}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{whose}\:\mathrm{domain}\:\mathrm{is}\:\mathbb{R} \\ $$$$\mathrm{satisfies}\:{f}\left({x}\right)={f}\left({x}+\mathrm{2}\right).\:\mathrm{When}\:{x}\:\in\:\left(\mathrm{0},\:\mathrm{1}\right), \\ $$$${f}\left({x}\right)=−\mathrm{2}{x}^{\mathrm{2}} +{ax}−\mathrm{2}. \\ $$$$\mathrm{If}\:{f}\:\mathrm{has}\:\mathrm{2023}\:\mathrm{zeros}\:\mathrm{in}\:\left[\mathrm{0},\:\mathrm{1011}\right].\:\mathrm{Then}\:\mathrm{the} \\ $$$$\mathrm{range}\:\mathrm{of}\:{a}\:\mathrm{can}\:\mathrm{be}\:? \\ $$$$\mathrm{A}.\:\left[−\mathrm{6},\:−\mathrm{2}\sqrt{\mathrm{2}}\right]\:\:\:\:\:\:\:\:\mathrm{B}.\:\left[−\mathrm{4},\:−\mathrm{2}\sqrt{\mathrm{2}}\right] \\ $$$$\mathrm{C}.\:\left[−\mathrm{8},\:−\mathrm{6}\right]\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{D}.\:\left[−\mathrm{6},\:−\mathrm{4}\right] \\ $$

Question Number 184869 Answers: 1 Comments: 0

Question Number 184866 Answers: 1 Comments: 1

Question Number 184861 Answers: 0 Comments: 0

x ∈ [−0,5 ; 0,5] find the product of all x′s 3(cos^2 πx + sinπy) + 2 = 9 + 3 ∣sinπx ∙ sinπy∣−sinπy

$$\mathrm{x}\:\in\:\left[−\mathrm{0},\mathrm{5}\:\:;\:\:\mathrm{0},\mathrm{5}\right] \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\:\boldsymbol{\mathrm{x}}'\mathrm{s} \\ $$$$\mathrm{3}\left(\mathrm{cos}^{\mathrm{2}} \pi\mathrm{x}\:+\:\mathrm{sin}\pi\mathrm{y}\right)\:+\:\mathrm{2}\:=\:\mathrm{9}\:+\:\mathrm{3}\:\mid\mathrm{sin}\pi\mathrm{x}\:\centerdot\:\mathrm{sin}\pi\mathrm{y}\mid−\mathrm{sin}\pi\mathrm{y} \\ $$

Question Number 184859 Answers: 2 Comments: 0

Lim_( x→∞) x^( 4) ( 1− cos (1− cos((2/x))))=?

$$ \\ $$$$\:\mathrm{Lim}_{\:{x}\rightarrow\infty} \:{x}^{\:\mathrm{4}} \:\left(\:\mathrm{1}−\:{cos}\:\left(\mathrm{1}−\:{cos}\left(\frac{\mathrm{2}}{{x}}\right)\right)\right)=? \\ $$$$ \\ $$

Question Number 184858 Answers: 1 Comments: 0

Σ_(n=o) ^(+oo) (((−1)^n x^(2n+1) )/(4n^2 −1))

$$\underset{{n}={o}} {\overset{+{oo}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} {x}^{\mathrm{2}{n}+\mathrm{1}} }{\mathrm{4}{n}^{\mathrm{2}} −\mathrm{1}} \\ $$

Question Number 184856 Answers: 0 Comments: 0

f(x)= (√( ∣ x_ ^ ∣ −∣ x−_ ^ ⌊ax⌋ ∣)) ; a ∈ [ 3 , 4 ) find : { (( D_( f) =? (domain ))),(( R_( f) =? (range ))) :}

$$ \\ $$$$\:\:\:\:\:{f}\left({x}\right)=\:\sqrt{\:\mid\:\underset{} {\overset{} {{x}}}\:\:\mid\:−\mid\:\:{x}\underset{} {\overset{} {−}}\lfloor{ax}\rfloor\:\mid} \\ $$$$\:\:\:\:\:\:;\:\:\:{a}\:\in\:\left[\:\mathrm{3}\:,\:\mathrm{4}\:\right) \\ $$$$\:\:\:\:\:{find}\::\:\:\:\begin{cases}{\:\:{D}_{\:{f}} \:=?\:\left({domain}\:\right)}\\{\:\:\mathcal{R}_{\:{f}} \:=?\:\left({range}\:\right)}\end{cases} \\ $$$$ \\ $$

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