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Question Number 53778 Answers: 0 Comments: 1

let U_n =(1/(nH_n )) with H_n =Σ_(k=1) ^n (1/k) study the convergence of Σ_(n≥1) U_n 2) study the convergence of Σ_(n≥1) U_n ^2

$${let}\:{U}_{{n}} =\frac{\mathrm{1}}{{nH}_{{n}} }\:\:\:\:{with}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$$${study}\:{the}\:{convergence}\:{of}\:\sum_{{n}\geqslant\mathrm{1}} \:{U}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{convergence}\:{of}\:\sum_{{n}\geqslant\mathrm{1}} {U}_{{n}} ^{\mathrm{2}} \\ $$

Question Number 53775 Answers: 0 Comments: 0

let A = (((2 1)),((−1 1)) ) 1) determine P inversible and D diagoanal in ordre to have A =PDP^(−1) 1) calculate A^n with n integr nstural 2) calculate e^(t A) with t ∈ R 3) calculate e^(−A) .

$${let}\:\:{A}\:=\:\begin{pmatrix}{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}}\\{−\mathrm{1}\:\:\:\:\:\:\:\:\mathrm{1}}\end{pmatrix} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:\:{P}\:{inversible}\:{and}\:{D}\:{diagoanal}\:{in}\:{ordre}\:{to}\:{have} \\ $$$${A}\:={PDP}^{−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}^{{n}} \:\:{with}\:{n}\:{integr}\:{nstural} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{e}^{{t}\:{A}} \:\:\:{with}\:{t}\:\in\:{R}\:\: \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:{e}^{−{A}} \:\:. \\ $$

Question Number 53770 Answers: 1 Comments: 0

Solve the equation: (he)^2 = she , where h, e and s are integers .

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{he}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{she}\:\:,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{where}\:\:\mathrm{h},\:\mathrm{e}\:\:\mathrm{and}\:\:\mathrm{s}\:\:\mathrm{are}\:\mathrm{integers}\:. \\ $$

Question Number 53769 Answers: 0 Comments: 4

Find maximum of ((xyz)/((x+a)(x+y)(y+z)(z+b))) .

$${Find}\:{maximum}\:{of} \\ $$$$\:\:\:\frac{{xyz}}{\left({x}+{a}\right)\left({x}+{y}\right)\left({y}+{z}\right)\left({z}+{b}\right)}\:\:. \\ $$

Question Number 53740 Answers: 1 Comments: 0

please help me to solve this sistem: { ((log_2 (x+2y)−log_3 (x−2y)=2)),((x^2 −4y^2 =4)) :}

$${please}\:{help}\:{me}\:{to}\:{solve}\:{this}\: \\ $$$${sistem}: \\ $$$$\begin{cases}{{log}_{\mathrm{2}} \left({x}+\mathrm{2}{y}\right)−{log}_{\mathrm{3}} \left({x}−\mathrm{2}{y}\right)=\mathrm{2}}\\{{x}^{\mathrm{2}} −\mathrm{4}{y}^{\mathrm{2}} =\mathrm{4}}\end{cases} \\ $$

Question Number 53736 Answers: 2 Comments: 4

(√(32+10(√7)))+(√(32−10(√7)))=?

$$\sqrt{\mathrm{32}+\mathrm{10}\sqrt{\mathrm{7}}}+\sqrt{\mathrm{32}−\mathrm{10}\sqrt{\mathrm{7}}}=? \\ $$

Question Number 53732 Answers: 1 Comments: 7

f(x)=(((x+a)(x+b))/((x−a)(x−b))) Find minimum and maximum.

$${f}\left({x}\right)=\frac{\left({x}+{a}\right)\left({x}+{b}\right)}{\left({x}−{a}\right)\left({x}−{b}\right)} \\ $$$${Find}\:{minimum}\:{and}\:{maximum}. \\ $$

Question Number 53755 Answers: 1 Comments: 9

Question Number 53720 Answers: 0 Comments: 1

Question Number 53709 Answers: 0 Comments: 1

A supermarket pays its sales personnel on a weekly basis. At the end of each week, each sales person receives a basic weekly wage plus bonus, which varies directly as the number of complete weeks that particular person has worked in the shop. At the end of her fourth week a sales girl received a pay packet containing $2060. Six weeks later her pay had jumped to $2150. Find the exact relation for determining how much the shop′s personnel are paid every week.

$$\mathrm{A}\:\mathrm{supermarket}\:\mathrm{pays}\:\mathrm{its}\:\mathrm{sales}\:\mathrm{personnel} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{weekly}\:\mathrm{basis}.\:\mathrm{At}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{each}\:\mathrm{week}, \\ $$$$\mathrm{each}\:\mathrm{sales}\:\mathrm{person}\:\mathrm{receives}\:\mathrm{a}\:\mathrm{basic}\: \\ $$$$\mathrm{weekly}\:\mathrm{wage}\:\mathrm{plus}\:\mathrm{bonus},\:\mathrm{which}\:\mathrm{varies} \\ $$$$\mathrm{directly}\:\mathrm{as}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{complete} \\ $$$$\mathrm{weeks}\:\mathrm{that}\:\mathrm{particular}\:\mathrm{person}\:\mathrm{has}\: \\ $$$$\mathrm{worked}\:\mathrm{in}\:\mathrm{the}\:\mathrm{shop}.\:\mathrm{At}\:\mathrm{the}\:\mathrm{end}\:\mathrm{of}\:\mathrm{her} \\ $$$$\mathrm{fourth}\:\mathrm{week}\:\mathrm{a}\:\mathrm{sales}\:\mathrm{girl}\:\mathrm{received}\:\mathrm{a}\:\mathrm{pay} \\ $$$$\mathrm{packet}\:\mathrm{containing}\:\$\mathrm{2060}.\:\mathrm{Six}\:\mathrm{weeks} \\ $$$$\mathrm{later}\:\mathrm{her}\:\mathrm{pay}\:\mathrm{had}\:\mathrm{jumped}\:\mathrm{to}\:\$\mathrm{2150}.\: \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{relation}\:\mathrm{for}\:\mathrm{determining} \\ $$$$\mathrm{how}\:\mathrm{much}\:\mathrm{the}\:\mathrm{shop}'\mathrm{s}\:\mathrm{personnel}\:\mathrm{are}\: \\ $$$$\mathrm{paid}\:\mathrm{every}\:\mathrm{week}. \\ $$

Question Number 53697 Answers: 0 Comments: 1

The solution of the equation ∫_(log 2) ^x (1/(√(e^x −1))) dx= (π/6) is given by

$$\mathrm{The}\:\mathrm{solution}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\: \\ $$$$\underset{\mathrm{log}\:\mathrm{2}} {\overset{{x}} {\int}}\:\:\frac{\mathrm{1}}{\sqrt{{e}^{{x}} −\mathrm{1}}}\:{dx}=\:\frac{\pi}{\mathrm{6}}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by} \\ $$

Question Number 53696 Answers: 2 Comments: 2

Let I_n =∫_( 0) ^(π/4) tan^n x dx, (n>1 and n∈N), then

$$\mathrm{Let}\:{I}_{{n}} =\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\mathrm{tan}^{{n}} {x}\:{dx},\:\left({n}>\mathrm{1}\:\mathrm{and}\:{n}\in{N}\right),\:\mathrm{then} \\ $$

Question Number 53695 Answers: 1 Comments: 1

If ∫_( 0) ^∞ e^(−x^2 ) dx = (√(π/2)) , then ∫_( 0) ^∞ e^(−ax^2 ) dx, a > 0 is

$$\mathrm{If}\:\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:{e}^{−{x}^{\mathrm{2}} } {dx}\:=\:\sqrt{\frac{\pi}{\mathrm{2}}}\:,\:\mathrm{then}\:\underset{\:\mathrm{0}} {\overset{\infty} {\int}}\:{e}^{−{ax}^{\mathrm{2}} } {dx}, \\ $$$${a}\:>\:\mathrm{0}\:\:\mathrm{is} \\ $$

Question Number 53694 Answers: 1 Comments: 0

∫_( 0) ^(r π) sin^(2n) x dx =