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Question Number 66799    Answers: 0   Comments: 0

let S_n =Σ_(k=1) ^n (((−1)^k )/(√k)) find a equivalent of S_n when n→+∞

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\sqrt{{k}}}\:\:{find}\:{a}\:{equivalent}\:{of}\:{S}_{{n}} \:{when}\:{n}\rightarrow+\infty \\ $$

Question Number 66798    Answers: 0   Comments: 0

find S_n =Σ_(k=0) ^n k^2 (C_n ^k )^2

$${find}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}^{\mathrm{2}} \:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} \\ $$

Question Number 66797    Answers: 0   Comments: 0

find S_n =Σ_(k=0) ^n (C_n ^k )^3

$${find}\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\left({C}_{{n}} ^{{k}} \right)^{\mathrm{3}} \\ $$

Question Number 66796    Answers: 0   Comments: 2

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

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

Question Number 66795    Answers: 0   Comments: 3

let f(x) =e^(−x) ln(1+x^2 ) 1) calculate f^((n)) (0) 2) developp f at integr serie

$${let}\:{f}\left({x}\right)\:={e}^{−{x}} {ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 66794    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((cos(2arctan(2x)))/(9+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left(\mathrm{2}{arctan}\left(\mathrm{2}{x}\right)\right)}{\mathrm{9}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 66793    Answers: 0   Comments: 0

calculate ∫_0 ^1 cos(3arctanx)dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{3}{arctanx}\right){dx} \\ $$

Question Number 66792    Answers: 0   Comments: 1

calculate ∫_0 ^1 cos(2 arctan(x))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{cos}\left(\mathrm{2}\:{arctan}\left({x}\right)\right){dx} \\ $$

Question Number 66791    Answers: 0   Comments: 1

let f(x) =cos(2arctanx) 1) calculate f^((n)) (0) 2)developp f at integr serie

$${let}\:\:{f}\left({x}\right)\:={cos}\left(\mathrm{2}{arctanx}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$

Question Number 66790    Answers: 0   Comments: 0

find ∫_0 ^∞ (x/(sh(x)))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}}{{sh}\left({x}\right)}{dx} \\ $$

Question Number 66789    Answers: 0   Comments: 1

sove the (de) (1+2(√x))y^′ −(x+(√(x−1)))y =xsin(2x)

$${sove}\:{the}\:\left({de}\right)\:\:\:\left(\mathrm{1}+\mathrm{2}\sqrt{{x}}\right){y}^{'} −\left({x}+\sqrt{{x}−\mathrm{1}}\right){y}\:={xsin}\left(\mathrm{2}{x}\right) \\ $$

Question Number 66788    Answers: 0   Comments: 0

solve the (de) (2x+1)y^′ +(x^2 −1)y =x^3 e^(−x)

$${solve}\:{the}\:\left({de}\right)\:\:\:\:\:\left(\mathrm{2}{x}+\mathrm{1}\right){y}^{'} \:\:\:+\left({x}^{\mathrm{2}} −\mathrm{1}\right){y}\:={x}^{\mathrm{3}} {e}^{−{x}} \\ $$

Question Number 66787    Answers: 0   Comments: 0

find the value of ∫_0 ^∞ (x^2 /(ch(x)))dx

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

Question Number 66786    Answers: 0   Comments: 1

find the value of ∫_0 ^∞ (x/(ch(x)))dx

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

Question Number 66778    Answers: 1   Comments: 4

(1/1)+(1/2)−(2/3)+(1/4)+(1/5)−(2/6)+(1/7)+(1/8)−(2/9)+(1/(10))+(1/(11))−(2/(12))+∙∙∙=

$$\frac{\mathrm{1}}{\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{2}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}}−\frac{\mathrm{2}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{7}}+\frac{\mathrm{1}}{\mathrm{8}}−\frac{\mathrm{2}}{\mathrm{9}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{11}}−\frac{\mathrm{2}}{\mathrm{12}}+\centerdot\centerdot\centerdot= \\ $$

Question Number 66769    Answers: 1   Comments: 2

simplify ((x+4)/(x−4))−((5x+20)/(x^2 −16))

$${simplify} \\ $$$$\frac{{x}+\mathrm{4}}{{x}−\mathrm{4}}−\frac{\mathrm{5}{x}+\mathrm{20}}{{x}^{\mathrm{2}} −\mathrm{16}} \\ $$

Question Number 66768    Answers: 0   Comments: 1

without using mathematical tables evaluate ((sin 60.tan 30.cos 60+sin 30.cos 45.sin 45)/(sin 90.cos 45.sin 45−sin 60.cos 30.sin 30))

$${without}\:{using}\:{mathematical}\:{tables} \\ $$$${evaluate} \\ $$$$\frac{\mathrm{sin}\:\mathrm{60}.\mathrm{tan}\:\mathrm{30}.\mathrm{cos}\:\mathrm{60}+\mathrm{sin}\:\mathrm{30}.\mathrm{cos}\:\mathrm{45}.\mathrm{sin}\:\mathrm{45}}{\mathrm{sin}\:\mathrm{90}.\mathrm{cos}\:\mathrm{45}.\mathrm{sin}\:\mathrm{45}−\mathrm{sin}\:\mathrm{60}.\mathrm{cos}\:\mathrm{30}.\mathrm{sin}\:\mathrm{30}} \\ $$

Question Number 66767    Answers: 0   Comments: 2

In a school there are 30 more boys than girls. One-quarter of the boys and two-thirds of the girls are boarders. If there are 255 boarders, find the number of students in the school.

$${In}\:{a}\:{school}\:{there}\:{are}\:\mathrm{30}\:{more}\:{boys} \\ $$$${than}\:{girls}.\:{One}-{quarter}\:{of}\:{the}\:{boys} \\ $$$${and}\:{two}-{thirds}\:{of}\:{the}\:{girls}\:{are}\:{boarders}. \\ $$$${If}\:{there}\:{are}\:\mathrm{255}\:{boarders},\:{find}\:{the} \\ $$$${number}\:{of}\:{students}\:{in}\:{the}\:{school}. \\ $$

Question Number 66765    Answers: 0   Comments: 2

Simba had 57 denomination notes which he deposited in his account. He had six times as many two-hundred shilling notes as one-thousand shilling notes and twice as many one-hundred shilling notes as two- hundred shilling notes. The rest were fifty shilling notes. If he deposited a total of sh 7750, find the number of fifty shilling notes he had.

$${Simba}\:{had}\:\mathrm{57}\:{denomination}\:{notes} \\ $$$${which}\:{he}\:{deposited}\:{in}\:{his}\:{account}. \\ $$$${He}\:{had}\:{six}\:{times}\:{as}\:{many}\:{two}-{hundred} \\ $$$${shilling}\:{notes}\:{as}\:{one}-{thousand}\: \\ $$$${shilling}\:{notes}\:{and}\:{twice}\:{as}\:{many}\: \\ $$$${one}-{hundred}\:{shilling}\:{notes}\:{as}\:{two}- \\ $$$${hundred}\:{shilling}\:{notes}.\:{The}\:{rest}\:{were} \\ $$$${fifty}\:{shilling}\:{notes}.\:{If}\:{he}\:{deposited} \\ $$$${a}\:{total}\:{of}\:{sh}\:\mathrm{7750},\:{find}\:{the}\:{number} \\ $$$${of}\:{fifty}\:{shilling}\:{notes}\:{he}\:{had}. \\ $$

Question Number 66760    Answers: 2   Comments: 2

By writing your answer in the form a^y simplify (3^(5x) ×5^(2x) ×3^(−x) ÷5^(−2x) )^(1/4)

$${By}\:{writing}\:{your}\:{answer}\:{in}\:{the} \\ $$$${form}\:{a}^{{y}} \:{simplify} \\ $$$$\left(\mathrm{3}^{\mathrm{5}{x}} ×\mathrm{5}^{\mathrm{2}{x}} ×\mathrm{3}^{−{x}} \boldsymbol{\div}\mathrm{5}^{−\mathrm{2}{x}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} \\ $$

Question Number 66759    Answers: 1   Comments: 1

Question Number 66756    Answers: 2   Comments: 0

solve the equations, x+y=17 xy−5x=32

$${solve}\:{the}\:{equations}, \\ $$$${x}+{y}=\mathrm{17} \\ $$$${xy}−\mathrm{5}{x}=\mathrm{32} \\ $$

Question Number 66755    Answers: 4   Comments: 0

solve the simultaneous equations: 3x-y=9 x^2 -xy=4

$${solve}\:{the}\:{simultaneous}\:{equations}: \\ $$$$\mathrm{3}{x}-{y}=\mathrm{9} \\ $$$${x}^{\mathrm{2}} -{xy}=\mathrm{4} \\ $$

Question Number 66751    Answers: 1   Comments: 0

A rostum is made by cutting off the upper part of a cone along a plane parallel to the base at (2/3) up the height. What fraction of the volume of the cone does the rostum represent?

$${A}\:{rostum}\:{is}\:{made}\:{by}\:{cutting}\:{off} \\ $$$${the}\:{upper}\:{part}\:{of}\:{a}\:{cone}\:{along}\:{a} \\ $$$${plane}\:{parallel}\:{to}\:{the}\:{base}\:{at}\:\frac{\mathrm{2}}{\mathrm{3}}\:{up}\: \\ $$$${the}\:{height}.\:{What}\:{fraction}\:{of}\:{the}\: \\ $$$${volume}\:{of}\:{the}\:{cone}\:{does}\:{the} \\ $$$${rostum}\:{represent}? \\ $$

Question Number 66746    Answers: 2   Comments: 0

A point T divides a line AB internally in the ratio 5:2. Given that A is (-4,10) and B is (10,3), find the coordinates of T.

$${A}\:{point}\:{T}\:\:{divides}\:{a}\:{line}\:{AB}\:{internally}\:{in}\:{the}\:{ratio}\:\mathrm{5}:\mathrm{2}.\:{Given}\:{that}\:{A}\:{is}\:\left(-\mathrm{4},\mathrm{10}\right)\:{and}\:{B}\:{is}\:\left(\mathrm{10},\mathrm{3}\right),\:{find}\:{the}\:{coordinates}\:{of}\:{T}. \\ $$

Question Number 66743    Answers: 0   Comments: 4

Each month a store owner can spend at most $100,000 on PC′s and laptops. A PC costs the store owner $1000 and a laptop costs him $1500. Each PC is sold for a profit of $400 while a laptop is sold for a profit of $700. The store owner estimates that at least 15 PC′s but no more than 80 are sold each month. He also estimates that the number of laptops sold is at most half the PC′s. How many PC′s and how many laptops should be sold in order to maximize the profit?

$$\mathrm{Each}\:\mathrm{month}\:\mathrm{a}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{can}\:\mathrm{spend}\:\mathrm{at} \\ $$$$\mathrm{most}\:\$\mathrm{100},\mathrm{000}\:\mathrm{on}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{laptops}.\:\mathrm{A} \\ $$$$\mathrm{PC}\:\mathrm{costs}\:\mathrm{the}\:\mathrm{store}\:\mathrm{owner}\:\$\mathrm{1000}\:\mathrm{and}\:\mathrm{a} \\ $$$$\mathrm{laptop}\:\mathrm{costs}\:\mathrm{him}\:\$\mathrm{1500}.\:\mathrm{Each}\:\mathrm{PC}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{400}\:\mathrm{while}\:\mathrm{a}\:\mathrm{laptop}\:\mathrm{is}\:\mathrm{sold} \\ $$$$\mathrm{for}\:\mathrm{a}\:\mathrm{profit}\:\mathrm{of}\:\$\mathrm{700}.\:\mathrm{The}\:\mathrm{store}\:\mathrm{owner}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{at}\:\mathrm{least}\:\mathrm{15}\:\mathrm{PC}'\mathrm{s}\:\mathrm{but}\:\mathrm{no}\:\mathrm{more}\:\mathrm{than} \\ $$$$\mathrm{80}\:\mathrm{are}\:\mathrm{sold}\:\mathrm{each}\:\mathrm{month}.\:\mathrm{He}\:\mathrm{also}\:\mathrm{estimates} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{laptops}\:\mathrm{sold}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most} \\ $$$$\mathrm{half}\:\mathrm{the}\:\mathrm{PC}'\mathrm{s}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{PC}'\mathrm{s}\:\mathrm{and}\:\mathrm{how} \\ $$$$\mathrm{many}\:\mathrm{laptops}\:\mathrm{should}\:\mathrm{be}\:\mathrm{sold}\:\mathrm{in}\:\mathrm{order}\:\mathrm{to} \\ $$$$\mathrm{maximize}\:\mathrm{the}\:\mathrm{profit}? \\ $$

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