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Question Number 159517    Answers: 1   Comments: 1

lim_(x→0) (1/x^2 )(e^(−2x) −((1+ax)/(1+bx))) a+b=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\left({e}^{−\mathrm{2}{x}} −\frac{\mathrm{1}+{ax}}{\mathrm{1}+{bx}}\right) \\ $$$${a}+{b}=? \\ $$

Question Number 159507    Answers: 2   Comments: 1

Question Number 159497    Answers: 0   Comments: 4

Prove that 1 + (1/4) + (1/9) + …+ (1/n^2 ) < 2 for n ∈ N .

$${Prove}\:\:{that}\:\: \\ $$$$\:\:\:\:\:\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{1}}{\mathrm{9}}\:+\:\ldots+\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\:\:<\:\:\mathrm{2} \\ $$$${for}\:\:{n}\:\in\:\:\mathbb{N}\:\:. \\ $$

Question Number 159494    Answers: 1   Comments: 0

Question Number 159493    Answers: 0   Comments: 0

Question Number 159491    Answers: 1   Comments: 0

Question Number 159490    Answers: 0   Comments: 1

Question Number 159488    Answers: 2   Comments: 0

Question Number 159480    Answers: 1   Comments: 1

Question Number 159479    Answers: 1   Comments: 0

Question Number 159477    Answers: 1   Comments: 0

Find: Ω =∫ (1/((x + (1/x))^2 )) dx

$$\mathrm{Find}:\:\:\:\Omega\:=\int\:\frac{\mathrm{1}}{\left(\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 159475    Answers: 0   Comments: 0

Question Number 159474    Answers: 1   Comments: 0

nice integral. prove that ∫^( ∞) _0 (( tan^( −1) (2x)+ tan^( −1) ((x/2) ))/(1+x^( 2) ))dx=(π^( 2) /4)

$$ \\ $$$$\:\:\:\:\:\:{nice}\:\:{integral}. \\ $$$$\:\:{prove}\:\:{that} \\ $$$$\underset{\mathrm{0}} {\int}^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \left(\mathrm{2}{x}\right)+\:{tan}^{\:−\mathrm{1}} \left(\frac{{x}}{\mathrm{2}}\:\right)}{\mathrm{1}+{x}^{\:\mathrm{2}} }{dx}=\frac{\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$

Question Number 159472    Answers: 1   Comments: 0

Question Number 159469    Answers: 1   Comments: 1

Question Number 159466    Answers: 2   Comments: 0

if tanA = (a/b) then prove that sinA = ± (a/( (√(a^ + b^ )))) please help..

$$\mathrm{if}\:{tanA}\:=\:\frac{{a}}{{b}} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:{sinA}\:=\:\pm\:\frac{{a}}{\:\sqrt{{a}^{ } \:+\:{b}^{ } }} \\ $$$$\mathrm{please}\:\mathrm{help}.. \\ $$

Question Number 159465    Answers: 1   Comments: 0

simplify ξ := Σ_(n=1) ^∞ ( (( 1)/(Σ_(k=1) ^n k^3 )) )=?

$$ \\ $$$$\:\:\:\:\:{simplify} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\xi\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\:\frac{\:\mathrm{1}}{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\mathrm{3}} }\:\right)=? \\ $$$$ \\ $$

Question Number 159453    Answers: 1   Comments: 1

x and y are real value and x+iy=(5/(−3+4i)) find x and y.(i=(√(−1)))

$$\mathrm{x}\:{and}\:\mathrm{y}\:{are}\:{real}\:{value}\:{and} \\ $$$$\:\mathrm{x}+\mathrm{iy}=\frac{\mathrm{5}}{−\mathrm{3}+\mathrm{4}{i}}\:{find}\:\mathrm{x}\:{and}\:\mathrm{y}.\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 159461    Answers: 0   Comments: 4

≺PRIME-BIRTHDAYS≻ Do you know ′Prime1611′?... No,no it′s not an ID of the forum- member.It is a person who was born on November 16, 0001.On his birthday astrologers formed a string from his birthdate in the way: ′ddmmyyyy′: ′16110001′ The astro- logers observed that number containing in the string: 16110001 is a prime number.Thus they gave him name ′Prime1611′ They also suggested that Mr Prime1611 should celebrate his birthday only when the number ddmmyyyy be prime number.Recently He celebrated his birthday on 16-11-2021 as the number 16112021 is prime. How many birthdays has he celebrated upto his recent birthday when he had celebrated his first birthday on 16-11-0001? You may use calculator also. Question connected with Q#159421

$$\:\:\:\:\:\:\:\:\:\:\prec\mathbb{PRIME}-\mathcal{BIRTHDAYS}\succ \\ $$$$\mathrm{Do}\:\mathrm{you}\:\mathrm{know}\:'\mathrm{Prime1611}'?... \\ $$$$\mathrm{No},\mathrm{no}\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{an}\:\mathrm{ID}\:\mathrm{of}\:\mathrm{the}\:\mathrm{forum}- \\ $$$$\mathrm{member}.\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{person}\:\mathrm{who}\:\mathrm{was}\:\mathrm{born} \\ $$$$\mathrm{on}\:\mathrm{November}\:\mathrm{16},\:\mathrm{0001}.\mathrm{On}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{astrologers}\:\mathrm{formed}\:\mathrm{a}\:\mathrm{string} \\ $$$$\mathrm{from}\:\mathrm{his}\:\mathrm{birthdate}\:\mathrm{in}\:\mathrm{the}\:\mathrm{way}: \\ $$$$'\mathrm{ddmmyyyy}':\:'\mathrm{16110001}'\:\mathrm{The}\:\mathrm{astro}- \\ $$$$\mathrm{logers}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{number}\:\mathrm{containing} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{string}:\:\mathrm{16110001}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime} \\ $$$$\mathrm{number}.\mathrm{Thus}\:\mathrm{they}\:\mathrm{gave}\:\mathrm{him}\:\mathrm{name} \\ $$$$'\mathrm{Prime1611}'\:\mathrm{They}\:\mathrm{also}\:\mathrm{suggested}\:\mathrm{that} \\ $$$$\mathrm{Mr}\:\mathrm{Prime1611}\:\mathrm{should}\:\mathrm{celebrate}\:\mathrm{his} \\ $$$$\mathrm{birthday}\:\mathrm{only}\:\mathrm{when}\:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{ddmmyyyy}\:\mathrm{be}\:\mathrm{prime}\:\mathrm{number}.\mathrm{Recently} \\ $$$$\mathrm{He}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{2021}\:\mathrm{as}\:\mathrm{the}\:\mathrm{number}\:\mathrm{16112021} \\ $$$$\mathrm{is}\:\mathrm{prime}. \\ $$$$\mathrm{How}\:\mathrm{many}\:\mathrm{birthdays}\:\mathrm{has}\:\mathrm{he}\:\mathrm{celebrated} \\ $$$$\mathrm{upto}\:\mathrm{his}\:\mathrm{recent}\:\mathrm{birthday}\:\mathrm{when}\:\mathrm{he} \\ $$$$\mathrm{had}\:\mathrm{celebrated}\:\mathrm{his}\:\mathrm{first}\:\mathrm{birthday}\:\mathrm{on} \\ $$$$\mathrm{16}-\mathrm{11}-\mathrm{0001}? \\ $$$$\mathrm{You}\:\mathrm{may}\:\mathrm{use}\:\mathrm{calculator}\:\mathrm{also}. \\ $$$$\mathrm{Question}\:\mathrm{connected}\:\mathrm{with}\:\mathrm{Q}#\mathrm{159421} \\ $$

Question Number 159447    Answers: 1   Comments: 1

#calculate# Ω := ∫_0 ^( 1) ∫_0 ^( 1) (( x^( (t/2)) −x^( t) )/(1 − x)) dx dt = ? −−−m.n−−−

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:#{calculate}# \\ $$$$\:\:\:\:\:\Omega\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\:{x}^{\:\frac{{t}}{\mathrm{2}}} −{x}^{\:{t}} }{\mathrm{1}\:−\:{x}}\:{dx}\:{dt}\:=\:? \\ $$$$\:\:\:\:\:\:\:\:\:−−−{m}.{n}−−− \\ $$

Question Number 159428    Answers: 1   Comments: 0

(dy/dx)=cos(x+y)+sin(x+y)

$$\frac{{dy}}{{dx}}={cos}\left({x}+{y}\right)+{sin}\left({x}+{y}\right) \\ $$

Question Number 159425    Answers: 0   Comments: 0

Question Number 159421    Answers: 0   Comments: 4

Question Number 159405    Answers: 1   Comments: 0

lim_(x→+0) (((Σ_(k=1) ^(2021) k^x )/(2021)))^(1/x) =?

$$\underset{\mathrm{x}\rightarrow+\mathrm{0}} {\mathrm{lim}}\left(\frac{\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{2021}} {\sum}}\mathrm{k}^{\mathrm{x}} }{\mathrm{2021}}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} =? \\ $$

Question Number 159403    Answers: 0   Comments: 1

U_(n+1) =(1/2)(u_n +(a/u_n )) with u_1 >0, a>0 Prove that (u_(n+1) /u_n )≤1

$${U}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{2}}\left({u}_{{n}} +\frac{{a}}{{u}_{{n}} }\right)\:{with}\:{u}_{\mathrm{1}} >\mathrm{0},\:\:{a}>\mathrm{0} \\ $$$${Prove}\:{that}\:\:\frac{{u}_{{n}+\mathrm{1}} }{{u}_{{n}} }\leqslant\mathrm{1} \\ $$

Question Number 159396    Answers: 1   Comments: 0

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