Question and Answers Forum

AllQuestion and Answers: Page 1763

Question Number 26571 Answers: 0 Comments: 1

let give I(x)= ∫_1 ^∝ ((t−E(t))/t^(x+1) )dt prove that ξ(x)= (x/(x−1)) −xI(x) then chow that (x−1)_(x−1^(+ ew) ) ξ(x)−−>1 we remind ξ(x) = Σ_(n≥1) (1/n^x ) and x>1

$${let}\:{give}\:\:{I}\left({x}\right)=\:\:\int_{\mathrm{1}} ^{\propto} \:\frac{{t}−{E}\left({t}\right)}{{t}^{{x}+\mathrm{1}} }{dt}\:\:\:{prove}\:{that} \\ $$$$\xi\left({x}\right)=\:\frac{{x}}{{x}−\mathrm{1}}\:−{xI}\left({x}\right)\:{then}\:{chow}\:{that}\:\left({x}−\mathrm{1}\right)_{{x}−\mathrm{1}^{+\:{ew}} } \xi\left({x}\right)−−>\mathrm{1} \\ $$$${we}\:{remind}\:\:\xi\left({x}\right)\:=\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{\mathrm{1}}{{n}^{{x}} }\:\:\:{and}\:\:{x}>\mathrm{1} \\ $$

Question Number 26570 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ e^(−px) /sinx/dx with p>0

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} {e}^{−{px}} /{sinx}/{dx}\:\:\:{with}\:{p}>\mathrm{0} \\ $$

Question Number 26569 Answers: 0 Comments: 1

find the value of ∫_0 ^(1 ) x E((1/x))dx

$${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}\:} {x}\:{E}\left(\frac{\mathrm{1}}{{x}}\right){dx}\: \\ $$

Question Number 26568 Answers: 0 Comments: 0

let give ξ(x)= Σ_(n=1) ^∝ (1/n^x ) prove that ξ(x)−_(x−>∝) 1∼2^(−x)

$${let}\:{give}\:\xi\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\propto} \frac{\mathrm{1}}{{n}^{{x}} }\:\:{prove}\:{that}\:\xi\left({x}\right)−_{{x}−>\propto} \mathrm{1}\sim\mathrm{2}^{−{x}} \\ $$

Question Number 26567 Answers: 1 Comments: 4

find the value of Σ_(n≥2) (((−)^n )/(n(n−1))) x^n

$${find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}\geqslant\mathrm{2}} \:\frac{\left(−\right)^{{n}} }{{n}\left({n}−\mathrm{1}\right)}\:{x}^{{n}} \\ $$

Question Number 26566 Answers: 0 Comments: 1

let give Γ(x)= ∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 prove that lim _(n−>∝) ∫_0 ^n (1−(t/n))^n t^(x−1) dt = Γ(x)

$${let}\:{give}\:\Gamma\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} {e}^{−{t}} {dt}\:{with}\:{x}>\mathrm{0}\:{prove}\:{that} \\ $$$$\:{lim}\:_{{n}−>\propto} \int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}} {t}^{{x}−\mathrm{1}} {dt}\:\:=\:\Gamma\left({x}\right) \\ $$

Question Number 26565 Answers: 0 Comments: 1

find the value of Σ_(n=0) ^∝ (((−1)^n )/(3n+1))

$${find}\:{the}\:{value}\:{of}\:\:\:\sum_{{n}=\mathrm{0}} ^{\propto} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\mathrm{3}{n}+\mathrm{1}} \\ $$

Question Number 26564 Answers: 0 Comments: 0

let give Γ(x)= ∫_0 ^∞ t^(x−1) e^(−t) dt and x>0(gamma euler function) prove that Γ(x) =lim_(n−>∝) (((n!) n^x )/(n(n+1)(n+2)...(n+x)))

$${let}\:{give}\:\Gamma\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} {t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt}\:\:\:{and}\:\:\:{x}>\mathrm{0}\left({gamma}\:{euler}\:{function}\right) \\ $$$${prove}\:{that}\:\:\Gamma\left({x}\right)\:\:={lim}_{{n}−>\propto} \:\frac{\left({n}!\right)\:{n}^{{x}} }{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)...\left({n}+{x}\right)} \\ $$

Question Number 26563 Answers: 1 Comments: 1

find the value of ∫_0 ^∞ ((1−cosx)/x^2 ) dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}−{cosx}}{{x}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 26584 Answers: 0 Comments: 4

∫_a ^x (x−t)^5 y(t)dt=4x^6 y(x)=...

$$\int_{{a}} ^{{x}} \left({x}−{t}\right)^{\mathrm{5}} {y}\left({t}\right){dt}=\mathrm{4}{x}^{\mathrm{6}} \\ $$$${y}\left({x}\right)=... \\ $$

Question Number 26576 Answers: 0 Comments: 4

Find all f : R→R such that f(x+f(x)+f(y))=f(y+f(x))+x+f(y)−f(f(y)) for all x, y ∈ R

$$\mathrm{Find}\:\mathrm{all}\:{f}\::\:\mathrm{R}\rightarrow\mathrm{R}\:\mathrm{such}\:\mathrm{that} \\ $$$${f}\left({x}+{f}\left({x}\right)+{f}\left({y}\right)\right)={f}\left({y}+{f}\left({x}\right)\right)+{x}+{f}\left({y}\right)−{f}\left({f}\left({y}\right)\right)\: \\ $$$$\mathrm{for}\:\mathrm{all}\:{x},\:\mathrm{y}\:\in\:\mathrm{R} \\ $$

Question Number 26575 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ e^(−[x]) sinxdx in that [x]=E(x)

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\left[{x}\right]} {sinxdx}\:\:\:{in}\:{that}\:\left[{x}\right]={E}\left({x}\right) \\ $$

Question Number 26574 Answers: 0 Comments: 1

study the nature of the serie Σ_(n=2) ^∝ ((cosn)/(√(n+(−1)^n ))) z^n

$${study}\:{the}\:{nature}\:{of}\:{the}\:{serie}\:\:\:\sum_{{n}=\mathrm{2}} ^{\propto} \:\:\frac{{cosn}}{\sqrt{{n}+\left(−\mathrm{1}\right)^{{n}} }}\:{z}^{{n}} \\ $$

Question Number 26561 Answers: 0 Comments: 0

Express sin(20)° in surd form.

$$\mathrm{Express}\:\:\mathrm{sin}\left(\mathrm{20}\right)°\:\:\mathrm{in}\:\mathrm{surd}\:\mathrm{form}. \\ $$

Question Number 26559 Answers: 0 Comments: 1

let put F(x)= ∫_0 ^∞ e^(−tx) ((sint)/t) dt with x≥0 we accept that F is class C^1 on [0,∝[ calculate (∂F/∂x) and find F(x) then find the value of ∫_0 ^∞ ((sint)/t) dt

$${let}\:{put}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{tx}} \:\frac{{sint}}{{t}}\:{dt}\:\:\:{with}\:\:{x}\geqslant\mathrm{0} \\ $$$${we}\:{accept}\:{that}\:{F}\:{is}\:{class}\:{C}^{\mathrm{1}} \:{on}\:\left[\mathrm{0},\propto\left[\right.\right. \\ $$$${calculate}\:\:\frac{\partial{F}}{\partial{x}}\:\:{and}\:{find}\:{F}\left({x}\right) \\ $$$${then}\:\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sint}}{{t}}\:{dt} \\ $$

Question Number 26558 Answers: 0 Comments: 1

find the value of ∫_0 ^∞ ((sinx)/(x(1+x^2 )))dx

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{sinx}}{{x}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{dx} \\ $$

Question Number 26557 Answers: 0 Comments: 0