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Question Number 40047    Answers: 0   Comments: 0

let S_n = Σ_(k=0) ^n (((−1)^k )/(2k+1)) 1) give S_n interms of H_n 2)find lim_(n→+∞) S_n 3) let W_n = Σ_(k=0) ^n (((−1)^k )/(4k^2 −1)) find W_n interms of H_n calculate lim_(n→+∞) W_n

$${let}\:{S}_{{n}} \:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{S}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} {S}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{W}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{4}{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$${find}\:{W}_{{n}} \:\:{interms}\:{of}\:{H}_{{n}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{W}_{{n}} \\ $$

Question Number 40046    Answers: 0   Comments: 2

let S_n = Σ_(k=2) ^n (((−1)^k )/(k^2 −1)) 1) calculate S_n interms of H_n ( H_n =Σ_(k=1) ^n (1/k)) 2) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{S}_{{n}} \:\:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left(\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$$$ \\ $$

Question Number 40044    Answers: 0   Comments: 2

let f(t) = ∫_0 ^(π/2) ln( cosx +t sinx) 1) calculate f(0) 2) calculate f^′ (t) then find a simple form of f(t) 3) calculate ∫_0 ^(π/2) ln(cosx +2 sinx)dx 4) calculate ∫_0 ^(π/2) ln((√3)cosx +sinx)dx

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\:{cosx}\:+{t}\:{sinx}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({t}\right)\:{then}\:{find}\:\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cosx}\:+\mathrm{2}\:{sinx}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\sqrt{\mathrm{3}}{cosx}\:+{sinx}\right){dx} \\ $$

Question Number 40043    Answers: 1   Comments: 0

find the value of ∫_(−1) ^(+∞) (√(x+1))e^(−x) dx

$${find}\:\:{the}\:{value}\:{of}\:\:\int_{−\mathrm{1}} ^{+\infty} \:\:\sqrt{{x}+\mathrm{1}}{e}^{−{x}} \:{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 40042    Answers: 0   Comments: 0

1) find the roots of p(x)=(1+ix +x^2 )^n −(1−ix+x^2 )^n with n integr natural 2) factorize p(x) inside C(x) 3) give p(x) at form Σ a_p x^p

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:\: \\ $$$${p}\left({x}\right)=\left(\mathrm{1}+{ix}\:+{x}^{\mathrm{2}} \right)^{{n}} −\left(\mathrm{1}−{ix}+{x}^{\mathrm{2}} \right)^{{n}} \:{with}\:{n}\:{integr} \\ $$$${natural} \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:\:{C}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{p}\left({x}\right)\:{at}\:{form}\:\:\Sigma\:{a}_{{p}} {x}^{{p}} \\ $$

Question Number 40040    Answers: 0   Comments: 1

let A_n = ∫_0 ^n e^(−n( x+2−[x])) dx with n integr natural 1) calculate A_n 2) find lim_(n→+∞) A_n 3) study the convergence of Σ_n A_n

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{e}^{−{n}\left(\:{x}+\mathrm{2}−\left[{x}\right]\right)} {dx}\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:\:\:\sum_{{n}} {A}_{{n}} \\ $$

Question Number 40037    Answers: 1   Comments: 0

Question Number 40031    Answers: 1   Comments: 1

Question Number 40023    Answers: 2   Comments: 0

Question Number 40022    Answers: 0   Comments: 2

To the developer Tinku Tara: Dear sir, can you please make it possible again that a posted image can be updated (changed) through the “edit post” function. This was possible in the past. But it is now not possible for unknown reasons. Thank you!

$${To}\:{the}\:{developer}\:{Tinku}\:{Tara}: \\ $$$${Dear}\:{sir}, \\ $$$${can}\:{you}\:{please}\:{make}\:{it}\:{possible}\:{again} \\ $$$${that}\:{a}\:{posted}\:{image}\:{can}\:{be}\:{updated} \\ $$$$\left({changed}\right)\:{through}\:{the}\:``{edit}\:{post}'' \\ $$$${function}. \\ $$$${This}\:{was}\:{possible}\:{in}\:{the}\:{past}.\:{But}\:{it}\:{is} \\ $$$${now}\:{not}\:{possible}\:{for}\:{unknown}\:{reasons}. \\ $$$${Thank}\:{you}! \\ $$

Question Number 40012    Answers: 1   Comments: 0

Question Number 40008    Answers: 1   Comments: 1

1) find ∫ (dx/((x+1)(√x) +x(√(x+1)))) .

$$\left.\mathrm{1}\right)\:{find}\:\:\int\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}}\:\:+{x}\sqrt{{x}+\mathrm{1}}}\:. \\ $$

Question Number 40007    Answers: 0   Comments: 1

find thevalue of ∫_0 ^∞ (dx/(1 +x^6 )) by using the value of ∫_(−∞) ^(+∞) (dx/(x−z)) with z ∈ C and Im(z)≠0

$${find}\:{thevalue}\:{of}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\mathrm{1}\:+{x}^{\mathrm{6}} }\:\:{by}\:{using}\:{the} \\ $$$${value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\frac{{dx}}{{x}−{z}}\:{with}\:{z}\:\in\:{C}\:{and}\:{Im}\left({z}\right)\neq\mathrm{0} \\ $$$$ \\ $$

Question Number 39995    Answers: 2   Comments: 0

Question Number 39993    Answers: 2   Comments: 0

Question Number 39985    Answers: 1   Comments: 0

Question Number 39983    Answers: 1   Comments: 0

To developer,Is there any way of getting back your previous account if you lost your password?

$${To}\:{developer},{Is}\:{there}\:{any}\:{way}\:{of} \\ $$$${getting}\:{back}\:{your}\:{previous}\:{account} \\ $$$${if}\:{you}\:{lost}\:{your}\:{password}? \\ $$

Question Number 39975    Answers: 0   Comments: 4

lim_(x→∞) (((∫_( 0) ^x e^x dx)^2 )/(∫_( 0) ^x e^(2x^2 ) dx)) =

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\underset{\:\mathrm{0}} {\overset{{x}} {\int}}\:\:{e}^{{x}} \:{dx}\right)^{\mathrm{2}} }{\underset{\:\mathrm{0}} {\overset{{x}} {\int}}\:\:{e}^{\mathrm{2}{x}^{\mathrm{2}} } \:{dx}}\:\:= \\ $$

Question Number 39971    Answers: 3   Comments: 0

a>0,b>0, What is the minimum value of ((b^2 +2)/(a+b))+(a^2 /(ab+1)) ?

$${a}>\mathrm{0},{b}>\mathrm{0}, \\ $$$${What}\:{is}\:{the}\:{minimum}\:{value}\:{of} \\ $$$$\frac{{b}^{\mathrm{2}} +\mathrm{2}}{{a}+{b}}+\frac{{a}^{\mathrm{2}} }{{ab}+\mathrm{1}}\:\:\:? \\ $$

Question Number 39970    Answers: 1   Comments: 0

Question Number 39955    Answers: 1   Comments: 2

Question Number 39967    Answers: 0   Comments: 4

1) decompose inside C(x) the fraction F(x)= (3/(4+x^4 )) 2) find ∫_(−∞) ^(+∞) (dx/(x−z)) with z from C 3) find the value of ∫_(−∞) ^(+∞) ((3dx)/(4+x^4 )) .

$$\left.\mathrm{1}\right)\:{decompose}\:{inside}\:{C}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\frac{\mathrm{3}}{\mathrm{4}+{x}^{\mathrm{4}} } \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{{x}−{z}}\:\:{with}\:{z}\:{from}\:{C} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{\mathrm{3}{dx}}{\mathrm{4}+{x}^{\mathrm{4}} }\:. \\ $$

Question Number 39929    Answers: 1   Comments: 0

Question Number 39928    Answers: 0   Comments: 1

what numbers is that of which the third Part exceeds the fifth part by 4

$${what}\:{numbers}\:{is}\:{that}\:{of}\:{which}\:{the}\:{third}\:{Part}\:{exceeds}\:{the}\:{fifth}\:{part}\:{by}\:\mathrm{4} \\ $$

Question Number 39921    Answers: 4   Comments: 0

Question Number 39930    Answers: 1   Comments: 0

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