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Question Number 31314 Answers: 0 Comments: 0

let x={(1/n)}_(n=1) ^∞ and y={(1/(n+1))}_(n=1) ^∞ be a sequence of real numbers and l_(2 ) ={x=(x_1 ,x_2 ,x_3 ,...):Σ_(n=1) ^∞ ∣xi∣^2 <∞} a linear space. (1) verify that x and y are in l_2 . (2) compute the inner product of x and y on l_2 please help me solve this question.

$${let}\:{x}=\left\{\frac{\mathrm{1}}{{n}}\right\}_{{n}=\mathrm{1}} ^{\infty} {and}\:{y}=\left\{\frac{\mathrm{1}}{{n}+\mathrm{1}}\right\}_{{n}=\mathrm{1}} ^{\infty} {be}\: \\ $$$${a}\:{sequence}\:{of}\:{real}\:{numbers}\:{and} \\ $$$${l}_{\mathrm{2}\:} =\left\{{x}=\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,{x}_{\mathrm{3}} ,...\right):\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mid{xi}\mid^{\mathrm{2}} <\infty\right\} \\ $$$${a}\:{linear}\:{space}.\: \\ $$$$\left(\mathrm{1}\right)\:{verify}\:{that}\:{x}\:{and}\:{y}\:{are}\:{in}\:{l}_{\mathrm{2}} . \\ $$$$\left(\mathrm{2}\right)\:{compute}\:{the}\:{inner}\:{product}\:{of}\:{x}\: \\ $$$${and}\:{y}\:{on}\:{l}_{\mathrm{2}} \\ $$$$\boldsymbol{{please}}\:\boldsymbol{{help}}\:\boldsymbol{{me}}\:\boldsymbol{{solve}}\:\boldsymbol{{this}}\: \\ $$$$\boldsymbol{{que}}{s}\boldsymbol{{tion}}. \\ $$

Question Number 31306 Answers: 1 Comments: 0

Complete the square in y^2 +8y+9k and hence find the value of k that makes it a perfect square.

$$\mathrm{Complete}\:\mathrm{the}\:\mathrm{square}\:\mathrm{in}\:\mathrm{y}^{\mathrm{2}} \:+\mathrm{8y}+\mathrm{9k}\:\mathrm{and}\:\mathrm{hence}\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{k}\:\mathrm{that}\:\mathrm{makes}\:\mathrm{it}\:\mathrm{a}\:\mathrm{perfect}\:\mathrm{square}. \\ $$

Question Number 31296 Answers: 0 Comments: 11

find ∫_0 ^(+∞) (dx/((1+x^2 )^n )) with n integr and n≥1 .

$${find}\:\:\int_{\mathrm{0}} ^{+\infty} \:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1}\:. \\ $$

Question Number 31295 Answers: 0 Comments: 0

Question Number 31289 Answers: 1 Comments: 0

Question Number 31287 Answers: 0 Comments: 0

Question Number 31286 Answers: 1 Comments: 1

Find all set of ordered triple/s (x,y,z), x,y,z∈ℜ, such that x−y=1−z 3(x^2 −y^2 )=5(1−z^2 ) 7(x^3 −y^3 )=19(1−z^3 ). Please show your solution.

$${Find}\:{all}\:{set}\:{of}\:{ordered}\:{triple}/{s}\:\left({x},{y},{z}\right),\:\:{x},{y},{z}\in\Re,\:{such}\:{that} \\ $$$${x}−{y}=\mathrm{1}−{z} \\ $$$$\mathrm{3}\left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)=\mathrm{5}\left(\mathrm{1}−{z}^{\mathrm{2}} \right) \\ $$$$\mathrm{7}\left({x}^{\mathrm{3}} −{y}^{\mathrm{3}} \right)=\mathrm{19}\left(\mathrm{1}−{z}^{\mathrm{3}} \right). \\ $$$${Please}\:{show}\:{your}\:{solution}. \\ $$

Question Number 31284 Answers: 0 Comments: 0

Question Number 31281 Answers: 1 Comments: 0

find three nos in AP whose product is equal to the square of their sum.

$${find}\:{three}\:{nos}\:{in}\:{AP}\:{whose}\:{product} \\ $$$${is}\:{equal}\:{to}\:{the}\:{square}\:{of}\:{their}\:{sum}. \\ $$

Question Number 31265 Answers: 0 Comments: 2

Question Number 31264 Answers: 0 Comments: 2

Find the shortest distance from the plane r^ .n^ =q to the sphere ∣r^ −r_0 ^ ∣=R .

$${Find}\:{the}\:{shortest}\:{distance}\:{from}\:{the} \\ $$$${plane}\:\bar {\boldsymbol{{r}}}.\bar {\boldsymbol{{n}}}=\boldsymbol{{q}}\:\:{to}\:{the}\:{sphere} \\ $$$$\:\mid\bar {\boldsymbol{{r}}}−\bar {\boldsymbol{{r}}}_{\mathrm{0}} \mid=\boldsymbol{{R}}\:\:. \\ $$

Question Number 31292 Answers: 1 Comments: 3

Question Number 31290 Answers: 1 Comments: 1

Question Number 31259 Answers: 0 Comments: 4

Question Number 31255 Answers: 1 Comments: 0

2n boys are randomly divided into two subgroups containing n boys each. The probability that the two tallest boys are in different groups is

$$\mathrm{2}{n}\:\mathrm{boys}\:\mathrm{are}\:\mathrm{randomly}\:\mathrm{divided}\:\mathrm{into}\:\mathrm{two} \\ $$$$\mathrm{subgroups}\:\mathrm{containing}\:{n}\:\mathrm{boys}\:\mathrm{each}.\:\mathrm{The} \\ $$$$\mathrm{probability}\:\mathrm{that}\:\mathrm{the}\:\mathrm{two}\:\mathrm{tallest}\:\mathrm{boys}\:\mathrm{are} \\ $$$$\mathrm{in}\:\mathrm{different}\:\mathrm{groups}\:\mathrm{is} \\ $$

Question Number 31249 Answers: 0 Comments: 10

2 lines through the point A(5, 1) are tangent to the circle x^2 + y^2 − 4x + 6y + 4 = 0 Find the equation of these 2 lines

$$\mathrm{2}\:\mathrm{lines}\:\mathrm{through}\:\mathrm{the}\:\mathrm{point}\:{A}\left(\mathrm{5},\:\mathrm{1}\right)\:\mathrm{are}\:\mathrm{tangent} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{circle}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:−\:\mathrm{4}{x}\:+\:\mathrm{6}{y}\:+\:\mathrm{4}\:=\:\mathrm{0} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{these}\:\mathrm{2}\:\mathrm{lines} \\ $$

Question Number 31246 Answers: 1 Comments: 2

without using lohpital find lim_(x→π/6) ((1−2sinx)/(cos 3x))

$$\mathrm{without}\:\mathrm{using}\:\mathrm{lohpital} \\ $$$$\mathrm{find} \\ $$$$\underset{\mathrm{x}\rightarrow\pi/\mathrm{6}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{2sinx}}{\mathrm{cos}\:\mathrm{3x}} \\ $$

Question Number 31237 Answers: 0 Comments: 3

Show that A−B = B′ ∩ A.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{A}−\mathrm{B}\:=\:\mathrm{B}'\:\cap\:\mathrm{A}. \\ $$

Question Number 31229 Answers: 0 Comments: 1

simplify A_n = C_n ^1 +2 C_n ^2 +3 C_n ^3 +... + n C_n ^n .

$${simplify}\:{A}_{{n}} =\:{C}_{{n}} ^{\mathrm{1}} \:\:+\mathrm{2}\:{C}_{{n}} ^{\mathrm{2}} \:+\mathrm{3}\:{C}_{{n}} ^{\mathrm{3}} \:+...\:+\:{n}\:{C}_{{n}} ^{{n}} \:. \\ $$

Question Number 31214 Answers: 2 Comments: 2

Question Number 31209 Answers: 1 Comments: 2

Question Number 31194 Answers: 2 Comments: 1

Find the remainder when x^(203) −1 is divided by x^4 −1.

$${Find}\:{the}\:{remainder}\:{when}\:{x}^{\mathrm{203}} −\mathrm{1} \\ $$$${is}\:{divided}\:{by}\:{x}^{\mathrm{4}} −\mathrm{1}. \\ $$

Question Number 31193 Answers: 1 Comments: 0

Question Number 31188 Answers: 0 Comments: 2

Question Number 31187 Answers: 0 Comments: 4

Question Number 31149 Answers: 1 Comments: 1

here is a question really troubling me. A cylindrical tube rolling down a slope of inclination θ moves a distance L in the time T. The equation relating these quantities is L(3+(a^2 /P))=QT^2 sin θ where a is the internal radius of the tube and P and Q are constants.What are the units of P and Q?

$${here}\:{is}\:{a}\:{question}\:{really}\:{troubling} \\ $$$${me}. \\ $$$$ \\ $$$${A}\:{cylindrical}\:{tube}\:{rolling}\:{down}\:{a} \\ $$$${slope}\:{of}\:{inclination}\:\theta\:{moves}\:{a} \\ $$$${distance}\:{L}\:{in}\:{the}\:{time}\:{T}.\:{The} \\ $$$${equation}\:{relating}\:{these}\:{quantities}\:{is} \\ $$$$ \\ $$$$\:\:\:{L}\left(\mathrm{3}+\frac{{a}^{\mathrm{2}} }{{P}}\right)={QT}^{\mathrm{2}} \mathrm{sin}\:\theta\:{where}\:{a}\:{is} \\ $$$${the}\:{internal}\:{radius}\:{of}\:{the}\:{tube}\:{and} \\ $$$${P}\:\:{and}\:{Q}\:{are}\:{constants}.{What}\:{are} \\ $$$${the}\:{units}\:{of}\:{P}\:{and}\:{Q}? \\ $$

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