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let give a sequence of real numbets positif (a_i )_(1≤i≤n) 1) prove that (Σ_(i=1) ^n a_i )^2 ≤ n Σ_(i=1) ^n a_i ^2 2)let put H_n =Σ_(k=1) ^n (1/k) and w_n = (H_n ^2 /n) prove that the sequence w_n is convergent .

$$\:{let}\:{give}\:{a}\:{sequence}\:{of}\:{real}\:{numbets}\:{positif} \\ $$$$\left({a}_{{i}} \right)_{\mathrm{1}\leqslant{i}\leqslant{n}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\left(\sum_{{i}=\mathrm{1}} ^{{n}} \:{a}_{{i}} \right)^{\mathrm{2}} \leqslant\:{n}\:\sum_{{i}=\mathrm{1}} ^{{n}} \:{a}_{{i}} ^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){let}\:{put}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\:\:{and}\:{w}_{{n}} =\:\frac{{H}_{{n}} ^{\mathrm{2}} }{{n}} \\ $$$${prove}\:{that}\:{the}\:{sequence}\:{w}_{{n}} \:{is}\:{convergent}\:. \\ $$

Question Number 33846 Answers: 0 Comments: 1

find radous of conbergence for theserie Σ_(n≥0) x^(n!) .

$${find}\:{radous}\:{of}\:{conbergence}\:{for}\:{theserie}\:\sum_{{n}\geqslant\mathrm{0}} {x}^{{n}!} .\: \\ $$

Question Number 33845 Answers: 0 Comments: 1

let I_n = ∫_0 ^1 ((arctan(1 +n))/(√(1+x^n ))) find lim_(n→+∞) I_n .

$${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{arctan}\left(\mathrm{1}\:+{n}\right)}{\sqrt{\mathrm{1}+{x}^{{n}} }}\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{I}_{{n}} \:. \\ $$

Question Number 33844 Answers: 0 Comments: 0

developp f(x)=e^(−cosx) at integr serie .

$${developp}\:{f}\left({x}\right)={e}^{−{cosx}} \:{at}\:{integr}\:{serie}\:. \\ $$

Question Number 33843 Answers: 0 Comments: 2

decompose inside R(x) the fraction F(x)= (1/((x+3)^n (x+1))) with n integr .

$${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$${F}\left({x}\right)=\:\:\frac{\mathrm{1}}{\left({x}+\mathrm{3}\right)^{{n}} \:\left({x}+\mathrm{1}\right)}\:{with}\:{n}\:{integr}\:. \\ $$

Question Number 33838 Answers: 1 Comments: 0

Find the values of k if ((x^2 +3x−4)/(5x−k)) may be capable of taking on all values when x is real.

$${Find}\:{the}\:{values}\:{of}\:{k}\:{if}\:\frac{{x}^{\mathrm{2}} +\mathrm{3}{x}−\mathrm{4}}{\mathrm{5}{x}−{k}}\: \\ $$$${may}\:{be}\:{capable}\:{of}\:{taking}\:{on}\:{all} \\ $$$${values}\:{when}\:{x}\:{is}\:{real}. \\ $$

Question Number 33837 Answers: 2 Comments: 0

find y if ∣((y−3)/(y+1))∣<2 i mean modulus by ∣

$${find}\:{y}\:{if}\:\mid\frac{{y}−\mathrm{3}}{{y}+\mathrm{1}}\mid<\mathrm{2} \\ $$$$ \\ $$$${i}\:{mean}\:{modulus}\:{by}\:\mid \\ $$

Question Number 33836 Answers: 2 Comments: 0

for what values of x if ((x(x−1))/(2x+3))>0

$${for}\:{what}\:{values}\:{of}\:{x}\:{if} \\ $$$$\frac{{x}\left({x}−\mathrm{1}\right)}{\mathrm{2}{x}+\mathrm{3}}>\mathrm{0} \\ $$

Question Number 33835 Answers: 0 Comments: 1

find the value of ∫_(−∞) ^(+∞) ((cos(πx))/((x^2 +1+i)^2 )) dx

$${find}\:{the}\:{value}\:{of}\:\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{cos}\left(\pi{x}\right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}+{i}\right)^{\mathrm{2}} }\:{dx} \\ $$

Question Number 33834 Answers: 0 Comments: 0

P(x)=x^n +a_(n−1) x^(n−1) +.... a_1 x+a_0 be a polynomial with all the real roots, prove that (n−1)a_(n−1) ^2 ≥ 2na_(n−2) .

$${P}\left({x}\right)={x}^{{n}} +{a}_{{n}−\mathrm{1}} {x}^{{n}−\mathrm{1}} +....\:{a}_{\mathrm{1}} {x}+{a}_{\mathrm{0}} \:\:\:{be}\:{a}\:{polynomial}\:{with}\:{all}\:{the}\:{real}\:{roots}, \\ $$$${prove}\:{that}\:\:\:\:\:\:\:\:\:\left({n}−\mathrm{1}\right){a}_{{n}−\mathrm{1}} ^{\mathrm{2}} \:\geqslant\:\mathrm{2}{na}_{{n}−\mathrm{2}} \:\:. \\ $$

Question Number 33825 Answers: 0 Comments: 0

Question Number 33822 Answers: 0 Comments: 1

The following table shows the distributuons of 100 families according to their expenditure per week. The mode is given to be 24. ∣((expenditure)/(Number of families))∣((10−20)/x)∣((20−30)/(27))∣((30−40)/y)∣((40−50)/(15))∣ (a) calculate the missing frequency (b)calculate the mean (c)calculate the median

Question Number 33819 Answers: 0 Comments: 0

given a=3i−4j+2k b=−2i+j−3k find (i) v=a+b (ii)v into position vector (iii)unit vector of v

$$\boldsymbol{\mathrm{given}}\:\:\:\boldsymbol{\mathrm{a}}=\mathrm{3}\boldsymbol{\mathrm{i}}−\mathrm{4}\boldsymbol{\mathrm{j}}+\mathrm{2}\boldsymbol{\mathrm{k}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{b}}=−\mathrm{2}\boldsymbol{\mathrm{i}}+\boldsymbol{\mathrm{j}}−\mathrm{3}\boldsymbol{\mathrm{k}} \\ $$$$\boldsymbol{\mathrm{find}}\:\:\:\left(\boldsymbol{\mathrm{i}}\right)\:\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{ii}}\right)\boldsymbol{\mathrm{v}}\:\boldsymbol{\mathrm{into}}\:\boldsymbol{\mathrm{position}}\:\boldsymbol{\mathrm{vector}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(\boldsymbol{\mathrm{iii}}\right)\boldsymbol{\mathrm{unit}}\:\boldsymbol{\mathrm{vector}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{v}} \\ $$

Question Number 33818 Answers: 1 Comments: 0

Let f:R → [ 1, ∞) be defined as f(x) = log_(10) ((√(3x^2 −4x+k+1)) +10 ). If f(x) is surjective , then find the value of k ?

$$\boldsymbol{{L}}{et}\:{f}:\boldsymbol{{R}}\:\rightarrow\:\left[\:\mathrm{1},\:\infty\right)\:{be}\:{defined}\:{as}\: \\ $$$${f}\left({x}\right)\:=\:\mathrm{log}_{\mathrm{10}} \:\left(\sqrt{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{4}{x}+\boldsymbol{{k}}+\mathrm{1}}\:+\mathrm{10}\:\right). \\ $$$$\boldsymbol{{I}}{f}\:{f}\left({x}\right)\:{is}\:\boldsymbol{{surjective}}\:,\:{then}\:{find} \\ $$$${the}\:{value}\:{of}\:\boldsymbol{{k}}\:? \\ $$

Question Number 33876 Answers: 0 Comments: 2

how can i write out 6 to thethird power divided by 2 to the fourth power pleasr help me write out this equation

$${how}\:{can}\:{i}\:{write}\:{out}\:\mathrm{6}\:{to}\:{thethird}\:{power}\:{divided}\:{by}\:\mathrm{2}\:{to}\:{the}\:{fourth}\:{power}\:{pleasr}\:{help}\:{me}\:{write}\:{out}\:{this}\:{equation}\: \\ $$

Question Number 33815 Answers: 1 Comments: 0

Let f:D → R be defined as f(x) = ((x^2 +2x+a)/(x^2 +4x+3a)) where D and R denote the domain of f and the set of all real numbers respectively. If f is ′′ surjective ′′ mapping then the range of a is ? a) 0≤a≤1 b) 0<a≤1 c) 0<a<1 d) 0≤a<1

$$\boldsymbol{{L}}{et}\:{f}:{D}\:\rightarrow\:\boldsymbol{{R}}\:{be}\:{defined}\:{as}\: \\ $$$${f}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+{a}}{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{3}{a}}\:{where}\:{D}\:{and}\:{R} \\ $$$${denote}\:{the}\:{domain}\:{of}\:\boldsymbol{{f}}\:{and}\:{the}\:{set} \\ $$$${of}\:{all}\:{real}\:{numbers}\:{respectively}. \\ $$$${If}\:{f}\:{is}\:''\:{surjective}\:''\:\:{mapping}\:{then} \\ $$$${the}\:{range}\:{of}\:\boldsymbol{{a}}\:{is}\:? \\ $$$$\left.{a}\right)\:\mathrm{0}\leqslant{a}\leqslant\mathrm{1} \\ $$$$\left.{b}\right)\:\mathrm{0}<{a}\leqslant\mathrm{1} \\ $$$$\left.{c}\right)\:\mathrm{0}<{a}<\mathrm{1} \\ $$$$\left.{d}\right)\:\mathrm{0}\leqslant{a}<\mathrm{1}\: \\ $$

Question Number 33805 Answers: 1 Comments: 0

The perimeter of a square and a rectangle is the same. The width of the rectangle is 6 cm and its area is 16 cm^2 less than the area of the square. Find the area of the square.

$$\mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{and}\:\mathrm{a}\:\mathrm{rectangle}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{same}.\:\mathrm{The}\:\mathrm{width}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rectangle}\:\mathrm{is}\:\mathrm{6}\:\mathrm{cm}\:\mathrm{and}\:\mathrm{its}\:\mathrm{area} \\ $$$$\mathrm{is}\:\mathrm{16}\:\mathrm{cm}^{\mathrm{2}} \:\mathrm{less}\:\mathrm{than}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}. \\ $$

Question Number 33800 Answers: 1 Comments: 0

Question Number 33787 Answers: 0 Comments: 0

lim_(n→∞) ((1/n) ∫_1 ^n n^(1/x) dx)

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{{n}}\:\underset{\mathrm{1}} {\overset{{n}} {\int}}\:{n}^{\frac{\mathrm{1}}{{x}}} \:{dx}\right) \\ $$

Question Number 33823 Answers: 1 Comments: 0

solve : I = ∫_0 ^π (((r−R cosθ) sin θ )/((R^(2 ) + r^2 − 2Rr cos θ)^(3/2) )) dθ for r < R and r > R respectively.

$$\:\:{solve}\::\: \\ $$$$\:{I}\:=\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\left({r}−{R}\:{cos}\theta\right)\:{sin}\:\theta\:}{\left({R}^{\mathrm{2}\:} +\:{r}^{\mathrm{2}} \:−\:\mathrm{2}{Rr}\:{cos}\:\theta\right)^{\mathrm{3}/\mathrm{2}} }\:{d}\theta \\ $$$${for}\:\:\:{r}\:<\:{R} \\ $$$${and}\:{r}\:>\:{R}\:\:{respectively}. \\ $$

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