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let u_k =1−(1−(1/2^k ))^(n−1) 1)prove that Σ u_k converges 2)let f(x)=1−(1−(1/2^x ))^(n−1) with x≥0 prove that ∀p∈N Σ_(k=1) ^(p+1) u_k ≤∫_0 ^(p+1) f(x)dx≤Σ_(k=0) ^p u_k

$${let}\:{u}_{{k}} =\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{k}} }\right)^{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\Sigma\:{u}_{{k}} {converges} \\ $$$$\left.\mathrm{2}\right){let}\:{f}\left({x}\right)=\mathrm{1}−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\right)^{{n}−\mathrm{1}} \:{with}\:{x}\geqslant\mathrm{0} \\ $$$${prove}\:{that}\:\forall{p}\in{N} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{p}+\mathrm{1}} \:{u}_{{k}} \:\leqslant\int_{\mathrm{0}} ^{{p}+\mathrm{1}} {f}\left({x}\right){dx}\leqslant\sum_{{k}=\mathrm{0}} ^{{p}} \:{u}_{{k}} \\ $$

Question Number 40892 Answers: 0 Comments: 0

let B(x,y) =∫_0 ^1 t^(x−1) (1−t)^(y−1) dt withx>0and y>0 prove that B(x,y)= ((Γ(x).Γ(y))/(Γ(x+y)))

$${let}\:{B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt} \\ $$$${withx}>\mathrm{0}{and}\:{y}>\mathrm{0}\:{prove}\:{that} \\ $$$${B}\left({x},{y}\right)=\:\frac{\Gamma\left({x}\right).\Gamma\left({y}\right)}{\Gamma\left({x}+{y}\right)} \\ $$

Question Number 40891 Answers: 0 Comments: 1

let x>0 and y>0 and B(x,y) =∫_0 ^1 t^(x−1) (1−t)^(y−1) dt 1)prove that B(x,y)=B(y,x) 2)B(x+1,y)=(x/y) B(x,y+1) 3)B(x+1,y)=(x/(x+y))B(x,y) 4)B(x,n+1)=((n!)/(x(x+1)....(x+n))) 5)B(n,p) = (1/((n+p−1)C_(n+p−2) ^(p−1) ))

$${let}\:{x}>\mathrm{0}\:{and}\:{y}>\mathrm{0}\:{and} \\ $$$${B}\left({x},{y}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {t}^{{x}−\mathrm{1}} \left(\mathrm{1}−{t}\right)^{{y}−\mathrm{1}} {dt} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:{B}\left({x},{y}\right)={B}\left({y},{x}\right) \\ $$$$\left.\mathrm{2}\right){B}\left({x}+\mathrm{1},{y}\right)=\frac{{x}}{{y}}\:{B}\left({x},{y}+\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right){B}\left({x}+\mathrm{1},{y}\right)=\frac{{x}}{{x}+{y}}{B}\left({x},{y}\right) \\ $$$$\left.\mathrm{4}\right){B}\left({x},{n}+\mathrm{1}\right)=\frac{{n}!}{{x}\left({x}+\mathrm{1}\right)....\left({x}+{n}\right)} \\ $$$$\left.\mathrm{5}\right){B}\left({n},{p}\right)\:=\:\frac{\mathrm{1}}{\left({n}+{p}−\mathrm{1}\right){C}_{{n}+{p}−\mathrm{2}} ^{{p}−\mathrm{1}} } \\ $$

Question Number 40890 Answers: 0 Comments: 2

1)calculate ∫_(1/(n+1)) ^(1/n) [(1/t)−[(1/t)]]dt 2)prove that ∫_0 ^1 [(1/t)−[(1/t)]]dt=1−γ γ is constant number of euler

$$\left.\mathrm{1}\right){calculate}\:\int_{\frac{\mathrm{1}}{{n}+\mathrm{1}}} ^{\frac{\mathrm{1}}{{n}}} \left[\frac{\mathrm{1}}{{t}}−\left[\frac{\mathrm{1}}{{t}}\right]\right]{dt} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left[\frac{\mathrm{1}}{{t}}−\left[\frac{\mathrm{1}}{{t}}\right]\right]{dt}=\mathrm{1}−\gamma \\ $$$$\gamma\:{is}\:{constant}\:{number}\:{of}\:{euler} \\ $$

Question Number 40889 Answers: 1 Comments: 0

prove?that ∫_0 ^1 ((1−(1−t)^n )/t)dt =Σ_(k=1) ^n (1/k)

$${prove}?{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{1}−\left(\mathrm{1}−{t}\right)^{{n}} }{{t}}{dt}\:=\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 40888 Answers: 0 Comments: 0

prove that ∀ξ ∈]0,π[ ∣∫_ξ ^π (((sint)/t))^n dt∣≤π(((sinξ)/ξ))^n n integr natural

$$\left.{prove}\:{that}\:\forall\xi\:\in\right]\mathrm{0},\pi\left[\right. \\ $$$$\mid\int_{\xi} ^{\pi} \left(\frac{{sint}}{{t}}\right)^{{n}} {dt}\mid\leqslant\pi\left(\frac{{sin}\xi}{\xi}\right)^{{n}} \:\:{n}\:{integr}\:{natural} \\ $$

Question Number 40887 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((tln(t))/(t^2 −1))dt

$${calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{tln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt} \\ $$

Question Number 40886 Answers: 0 Comments: 0

prove that ∫_0 ^1 ((t^(2p+1) ln(t))/(t^2 −1))dt =(π^2 /(24)) −(1/4)Σ_(k=1) ^p (1/k^2 )

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{t}^{\mathrm{2}{p}+\mathrm{1}} {ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{24}}\:−\frac{\mathrm{1}}{\mathrm{4}}\sum_{{k}=\mathrm{1}} ^{{p}} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} } \\ $$

Question Number 40885 Answers: 0 Comments: 1

prove that 1) ∫_0 ^1 ((t^p ln(t))/(t−1))dt =(π^2 /6) −Σ_(k=1) ^p (1/k^2 ) 2) ∫_0 ^1 ((t^(2p) ln(t))/(t^2 −1))dt =(π^2 /8) −Σ_(k=0) ^(p−1) (1/((2k+1)^2 ))

$${prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{t}^{{p}} {ln}\left({t}\right)}{{t}−\mathrm{1}}{dt}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:−\sum_{{k}=\mathrm{1}} ^{{p}} \:\frac{\mathrm{1}}{{k}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{t}^{\mathrm{2}{p}} {ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}\:=\frac{\pi^{\mathrm{2}} }{\mathrm{8}}\:−\sum_{{k}=\mathrm{0}} ^{{p}−\mathrm{1}} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{k}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$

Question Number 40884 Answers: 2 Comments: 0

1) fond ∫_0 ^1 ((ln(t))/(t^2 −1))dt 2) find ∫_0 ^1 ((ln(t))/(t^4 −1))dt

$$\left.\mathrm{1}\right)\:{fond}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ln}\left({t}\right)}{{t}^{\mathrm{2}} −\mathrm{1}}{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({t}\right)}{{t}^{\mathrm{4}} −\mathrm{1}}{dt} \\ $$

Question Number 40883 Answers: 1 Comments: 0

find ∫_0 ^∞ (t^p /(e^t −1))dt with p∈N^★

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{t}^{{p}} }{{e}^{{t}} −\mathrm{1}}{dt}\:{with}\:{p}\in{N}^{\bigstar} \\ $$

Question Number 40882 Answers: 0 Comments: 0

1)prove that ∀n≥2(n inyegr) x^(2n) −1=(x−1)(x+1)Π_(k=1) ^(n−1) (x^2 −2cos(((kπ)/n))x+1) 2)find the value of ∫_0 ^π ln(x^2 −2xcost +1)dt

$$\left.\mathrm{1}\right){prove}\:{that}\:\forall{n}\geqslant\mathrm{2}\left({n}\:{inyegr}\right) \\ $$$${x}^{\mathrm{2}{n}} −\mathrm{1}=\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \left({x}^{\mathrm{2}} \:−\mathrm{2}{cos}\left(\frac{{k}\pi}{{n}}\right){x}+\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcost}\:+\mathrm{1}\right){dt} \\ $$

Question Number 40880 Answers: 0 Comments: 0

prove that Σ_(k=n) ^∞ (1/k^α ) ∼ (1/((α−1)n^(α−1) ))with α>1

$${prove}\:{that}\:\sum_{{k}={n}} ^{\infty} \:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:\:\frac{\mathrm{1}}{\left(\alpha−\mathrm{1}\right){n}^{\alpha−\mathrm{1}} }{with}\:\alpha>\mathrm{1} \\ $$

Question Number 40878 Answers: 0 Comments: 0

let u_0 >0 and ∀n∈N u_(n+1) =u_n +(1/u_n ) 1) prove that (u_n )is increasing and lim u_n =+∞ 2)by consideringthe functionϕ(t)=(1/(2t+x)) prove that ∀n∈N Σ_(k=1) ^n (1/(2k+x)) ≤(1/2)ln(1+((2n)/x)) 3)find a equivalent of u_n (n→+∞)

$${let}\:{u}_{\mathrm{0}} >\mathrm{0}\:{and}\:\forall{n}\in{N} \\ $$$${u}_{{n}+\mathrm{1}} ={u}_{{n}} \:+\frac{\mathrm{1}}{{u}_{{n}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{increasing}\:{and}\:{lim}\:{u}_{{n}} \:=+\infty \\ $$$$\left.\mathrm{2}\right){by}\:{consideringthe}\:{function}\varphi\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}{t}+{x}} \\ $$$${prove}\:{that}\:\forall{n}\in{N}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{\mathrm{2}{k}+{x}}\:\leqslant\frac{\mathrm{1}}{\mathrm{2}}{ln}\left(\mathrm{1}+\frac{\mathrm{2}{n}}{{x}}\right) \\ $$$$\left.\mathrm{3}\right){find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \left({n}\rightarrow+\infty\right) \\ $$

Question Number 40876 Answers: 0 Comments: 0

prove by recurrence that Σ_(k=1) ^n k^4 =((n(n+1)(2n+1)(3n^2 +3n−1))/(30))

$${prove}\:{by}\:{recurrence}\:{that}\: \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{4}} =\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)\left(\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{3}{n}−\mathrm{1}\right)}{\mathrm{30}} \\ $$

Question Number 40875 Answers: 1 Comments: 0

2a sin(((25)/a)) − 51 = 0, find a

$$\mathrm{2a}\:\mathrm{sin}\left(\frac{\mathrm{25}}{\mathrm{a}}\right)\:−\:\mathrm{51}\:=\:\mathrm{0},\:\:\mathrm{find}\:\mathrm{a} \\ $$

Question Number 40874 Answers: 0 Comments: 0

Question Number 40873 Answers: 1 Comments: 0

If a^3 +b^3 =0, prove that log (a+b)=(1/2)(log a +log b +log 3) [given a+b≠0]

$${If}\:{a}^{\mathrm{3}} +{b}^{\mathrm{3}} =\mathrm{0},\:\:{prove}\:{that}\:\mathrm{log}\:\left({a}+{b}\right)=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{log}\:{a}\:+\mathrm{log}\:{b}\:+\mathrm{log}\:\mathrm{3}\right) \\ $$$$\left[{given}\:{a}+{b}\neq\mathrm{0}\right] \\ $$

Question Number 40872 Answers: 1 Comments: 2

If a^3 +b^3 =0, prove that log (a+b)=(1/2)(log a +log b +log 3) [given a+b≠0]

Question Number 40870 Answers: 1 Comments: 1

fnd ∫ (1+(1/x^2 ))arctan(x−(1/x))dx .

$${fnd}\:\:\int\:\:\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctan}\left({x}−\frac{\mathrm{1}}{{x}}\right){dx}\:. \\ $$

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