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Question Number 43003 Answers: 1 Comments: 0

let u_n = Σ_(1≤i<j≤n) (1/(√(ij))) 1) find a equivalent of u_n 2)calculate lim_(n→+∞) u_n

$${let}\:{u}_{{n}} =\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:\:\frac{\mathrm{1}}{\sqrt{{ij}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{equivalent}\:{of}\:{u}_{{n}} \\ $$$$\left.\mathrm{2}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 43001 Answers: 1 Comments: 0

prove that Π_(k=1) ^n (1+(1/k))>1+Σ_(k=1) ^n (1/k)

$${prove}\:{that}\:\prod_{{k}=\mathrm{1}} ^{{n}} \left(\mathrm{1}+\frac{\mathrm{1}}{{k}}\right)>\mathrm{1}+\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$

Question Number 43000 Answers: 2 Comments: 0

prove that (√(2+(√(2+....+(√2))))) =2cos((π/2^n ))

$${prove}\:{that}\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+....+\sqrt{\mathrm{2}}}}\:\:=\mathrm{2}{cos}\left(\frac{\pi}{\mathrm{2}^{{n}} }\right) \\ $$

Question Number 42999 Answers: 0 Comments: 1

let f(x)=(√x)+(1/(x−1)) 1) calculate f^((n)) (2) 2) if f(x) =Σ_(n=0) ^∞ a_n (x−2)^n find the sequence a_n

$${let}\:{f}\left({x}\right)=\sqrt{{x}}+\frac{\mathrm{1}}{{x}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{2}\right) \\ $$$$\left.\mathrm{2}\right)\:{if}\:{f}\left({x}\right)\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{a}_{{n}} \left({x}−\mathrm{2}\right)^{{n}} \:{find}\:{the} \\ $$$${sequence}\:{a}_{{n}} \\ $$

Question Number 42995 Answers: 0 Comments: 0

Dear Jr inter students use this firmulas Derivates d/dx constant(k)=0 d/dx x^n =n.x^(n−1) d/dx x^2 =2x d/dx (√x) =1/2(√x) d/dx e^x =e^x d/dx a^x =a^x loxa d/dx logx =1/x _ d dx 1/x =−1/x^2

Question Number 42994 Answers: 0 Comments: 5

∫(√(1+((cos x)/(4tan x))))dx=?

$$\int\sqrt{\mathrm{1}+\frac{\mathrm{cos}\:{x}}{\mathrm{4tan}\:{x}}}{dx}=? \\ $$

Question Number 42988 Answers: 0 Comments: 0

reduce this matrix [(2,3,4,1),(1,7,2,3),((−1),4,2,0),(0,1,1,0) ]

$${reduce}\:{this}\:{matrix}\begin{bmatrix}{\mathrm{2}}&{\mathrm{3}}&{\mathrm{4}}&{\mathrm{1}}\\{\mathrm{1}}&{\mathrm{7}}&{\mathrm{2}}&{\mathrm{3}}\\{−\mathrm{1}}&{\mathrm{4}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{0}}&{\mathrm{1}}&{\mathrm{1}}&{\mathrm{0}}\end{bmatrix} \\ $$

Question Number 42948 Answers: 1 Comments: 5

Question Number 42945 Answers: 2 Comments: 12

∫_0 ^( π/2) (dx/(√(sin x))) = ?

$$\int_{\mathrm{0}} ^{\:\:\pi/\mathrm{2}} \frac{{dx}}{\sqrt{\mathrm{sin}\:{x}}}\:=\:? \\ $$

Question Number 42942 Answers: 2 Comments: 0

Question Number 42934 Answers: 0 Comments: 2

Question Number 42933 Answers: 0 Comments: 1

Suppose that f and g are two functions such that lim_(x→a) g(x) = 0 and lim_(x→a) ((f(x))/(g(x))) exist. Prove that lim_(x→a) f(x) = 0

$$\mathrm{Suppose}\:\mathrm{that}\:{f}\:\mathrm{and}\:{g}\:\mathrm{are}\:\mathrm{two}\:\mathrm{functions}\:\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{g}\left({x}\right)\:=\:\mathrm{0}\:\:\:\:\mathrm{and}\:\:\:\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:\frac{{f}\left({x}\right)}{{g}\left({x}\right)}\:\:\:\mathrm{exist}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\:\mathrm{0} \\ $$

Question Number 42910 Answers: 1 Comments: 0

whatshouldbesubtructedfrom(2a+8b+90)toget(−3a+7b+16)?

$${whatshouldbesubtructedfrom}\left(\mathrm{2}{a}+\mathrm{8}{b}+\mathrm{90}\right){toget}\left(−\mathrm{3}{a}+\mathrm{7}{b}+\mathrm{16}\right)? \\ $$

Question Number 42909 Answers: 1 Comments: 0

How many pairs of (a, b, c, d) so that a! + b! + c! = d! which a, b, c, d ∈ positive integers .

$${How}\:\:{many}\:\:{pairs}\:{of}\:\:\left({a},\:{b},\:{c},\:{d}\right)\:\:{so}\:\:{that} \\ $$$$\:\:\:\:\:\:\:\:\:{a}!\:+\:\:{b}!\:\:+\:\:{c}!\:\:=\:\:{d}! \\ $$$${which}\:\:\:{a},\:{b},\:{c},\:{d}\:\:\in\:\:{positive}\:\:{integers}\:. \\ $$

Question Number 42908 Answers: 1 Comments: 0

Solve 21^a + 28^b = 35^c if a, b, and c are positive integers.

$${Solve}\: \\ $$$$\:\:\:\:\:\mathrm{21}^{{a}} \:+\:\mathrm{28}^{{b}} \:\:=\:\:\mathrm{35}^{{c}} \\ $$$${if} \\ $$$${a},\:{b},\:\:{and}\:\:{c}\:\:{are}\:\:{positive}\:\:{integers}. \\ $$

Question Number 42906 Answers: 3 Comments: 0

x^x +y^y =31 x+y = 5 Find x and y

$${x}^{{x}} +{y}^{{y}} =\mathrm{31} \\ $$$${x}+{y}\:=\:\mathrm{5} \\ $$$${Find}\:{x}\:{and}\:{y} \\ $$

Question Number 42897 Answers: 2 Comments: 1

Question Number 42895 Answers: 1 Comments: 1

Question Number 42878 Answers: 1 Comments: 0

Question Number 42870 Answers: 0 Comments: 0

let 0<x<1 and Γ(x) =∫_0 ^∞ t^(x−1) e^(−t) dt 1) prove that Γ(x).Γ(1−x) =(π/(sin(πx))) (compliments formulae) 2) calculate Γ(n) and Γ(n+(1/2)) with n from N.

$${let}\:\mathrm{0}<{x}<\mathrm{1}\:\:{and}\:\Gamma\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\: \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right).\Gamma\left(\mathrm{1}−{x}\right)\:=\frac{\pi}{{sin}\left(\pi{x}\right)}\:\:\:\left({compliments}\:{formulae}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\right)\:{and}\:\Gamma\left({n}+\frac{\mathrm{1}}{\mathrm{2}}\right)\:{with}\:{n}\:{from}\:{N}. \\ $$

Question Number 42866 Answers: 2 Comments: 0

Question Number 42863 Answers: 2 Comments: 7

Question Number 42861 Answers: 1 Comments: 0

14÷2(3+4)=?

$$\mathrm{14}\boldsymbol{\div}\mathrm{2}\left(\mathrm{3}+\mathrm{4}\right)=? \\ $$

Question Number 42826 Answers: 0 Comments: 7

solving ax^4 +bx^3 +cx^2 +dx+e=0 (a≠0, b, c, d, e)∈Q special cases (easy to solve) ax^4 +e=0 solve at^2 +e=0 ⇒ x=±(√t_(1, 2) ) ax^4 +cx^2 +e=0 solve at^2 +ct+e=0 ⇒ x=±(√t_(1, 2) ) always try all factors of ±e because a(x−α)(x−β)(x−γ)(x−δ)=ax^4 +...+αβγδ ⇒ e=αβγδ next we must find the nature of the solutions 4 real solutions 2 real & 2 complex solutions 4 complex solutions a, b, c, d, e ∈Q ⇒ complex solutions always in conjugated pairs draw the function or calculate some values to find the number of real solutions divide by a x^4 +px^3 +qx^2 +rx+s=0 [p=(b/a) q=(c/a) r=(d/a) s=(e/a)] I′ll soon post some cases I′ve been able to solve as comments

$$\mathrm{solving} \\ $$$${ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e}=\mathrm{0} \\ $$$$\left({a}\neq\mathrm{0},\:{b},\:{c},\:{d},\:{e}\right)\in\mathbb{Q} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{special}\:\mathrm{cases}\:\left(\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{4}} +{e}=\mathrm{0}\:\mathrm{solve}\:{at}^{\mathrm{2}} +{e}=\mathrm{0}\:\Rightarrow\:{x}=\pm\sqrt{{t}_{\mathrm{1},\:\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:{ax}^{\mathrm{4}} +{cx}^{\mathrm{2}} +{e}=\mathrm{0}\:\mathrm{solve}\:{at}^{\mathrm{2}} +{ct}+{e}=\mathrm{0}\:\Rightarrow\:{x}=\pm\sqrt{{t}_{\mathrm{1},\:\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{always}\:\mathrm{try}\:\mathrm{all}\:\mathrm{factors}\:\mathrm{of}\:\pm{e} \\ $$$$\mathrm{because}\:{a}\left({x}−\alpha\right)\left({x}−\beta\right)\left({x}−\gamma\right)\left({x}−\delta\right)={ax}^{\mathrm{4}} +...+\alpha\beta\gamma\delta \\ $$$$\Rightarrow\:{e}=\alpha\beta\gamma\delta \\ $$$$ \\ $$$$\mathrm{next}\:\mathrm{we}\:\mathrm{must}\:\mathrm{find}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solutions} \\ $$$$\mathrm{4}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$\mathrm{2}\:\mathrm{real}\:\&\:\mathrm{2}\:\mathrm{complex}\:\mathrm{solutions} \\ $$$$\mathrm{4}\:\mathrm{complex}\:\mathrm{solutions} \\ $$$${a},\:{b},\:{c},\:{d},\:{e}\:\in\mathbb{Q}\:\Rightarrow\:\mathrm{complex}\:\mathrm{solutions}\:\mathrm{always}\:\mathrm{in} \\ $$$$\mathrm{conjugated}\:\mathrm{pairs} \\ $$$$\mathrm{draw}\:\mathrm{the}\:\mathrm{function}\:\mathrm{or}\:\mathrm{calculate}\:\mathrm{some}\:\mathrm{values} \\ $$$$\mathrm{to}\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$ \\ $$$$\mathrm{divide}\:\mathrm{by}\:{a} \\ $$$${x}^{\mathrm{4}} +{px}^{\mathrm{3}} +{qx}^{\mathrm{2}} +{rx}+{s}=\mathrm{0} \\ $$$$\left[{p}=\frac{{b}}{{a}}\:\:{q}=\frac{{c}}{{a}}\:\:{r}=\frac{{d}}{{a}}\:\:{s}=\frac{{e}}{{a}}\right] \\ $$$$ \\ $$$$\mathrm{I}'\mathrm{ll}\:\mathrm{soon}\:\mathrm{post}\:\mathrm{some}\:\mathrm{cases}\:\mathrm{I}'\mathrm{ve}\:\mathrm{been}\:\mathrm{able}\:\mathrm{to}\:\mathrm{solve} \\ $$$$\mathrm{as}\:\mathrm{comments} \\ $$

Question Number 42823 Answers: 1 Comments: 1

Evaluate : ∫_(−5) ^( 5) x^2 [x+(1/2)]dx = ? where [.]= greatest integer function

$$\mathrm{Evaluate}\:: \\ $$$$\int_{−\mathrm{5}} ^{\:\mathrm{5}} \:{x}^{\mathrm{2}} \left[{x}+\frac{\mathrm{1}}{\mathrm{2}}\right]{dx}\:=\:\:? \\ $$$${where}\:\left[.\right]=\:{greatest}\:{integer}\:{function} \\ $$

Question Number 42812 Answers: 0 Comments: 1

study the convervence of ∫_1 ^(+∞) ((arctan(x−1))/((x^2 −1)^(4/3) )) dx

$${study}\:{the}\:{convervence}\:{of}\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{arctan}\left({x}−\mathrm{1}\right)}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\frac{\mathrm{4}}{\mathrm{3}}} }\:{dx} \\ $$

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