Question and Answers Forum

let S_n = Σ_(k=1) ^n (((−1)^k )/k) 1) calculate S_n interms of H_n 2) find lim_(n→+∞) S_n 3) let W_n = Σ_(1≤i<j≤n) (((−1)^(i+j) )/(i.j)) prove that (W_n ) is convergent and calculste its limit.

$${let}\:\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{S}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{W}_{{n}} =\:\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \frac{\left(−\mathrm{1}\right)^{{i}+{j}} }{{i}.{j}} \\ $$$${prove}\:{that}\:\left({W}_{{n}} \right)\:{is}\:{convergent}\:{and}\:{calculste}\:{its} \\ $$$${limit}. \\ $$

Question Number 40063 Answers: 1 Comments: 0

Question Number 40060 Answers: 0 Comments: 0

Mr John bought a four roomed house for 2520000bucks. these rooms are rented out to four students ,at 9000 buck per month for each room. a) find the rents collected at the end of each year.if each year 72000 bucks is spent on repairs. b) find the real annual income on the house

$${Mr}\:{John}\:{bought}\:{a}\:{four}\:{roomed} \\ $$$${house}\:{for}\:\mathrm{2520000}{bucks}. \\ $$$${these}\:{rooms}\:{are}\:{rented}\:{out} \\ $$$${to}\:{four}\:{students}\:,{at}\:\mathrm{9000}\:{buck} \\ $$$${per}\:{month}\:{for}\:{each}\:{room}. \\ $$$$\left.{a}\right)\:{find}\:{the}\:{rents}\:{collected}\:{at}\:{the}\:{end} \\ $$$${of}\:{each}\:{year}.{if}\:{each}\:{year}\:\mathrm{72000}\:{bucks} \\ $$$${is}\:{spent}\:{on}\:{repairs}. \\ $$$$\left.{b}\right)\:{find}\:{the}\:{real}\:{annual}\:{income}\:{on}\:{the}\:{house} \\ $$$$ \\ $$

Question Number 40058 Answers: 2 Comments: 0

Question Number 40057 Answers: 1 Comments: 0

Mike and Stev had 620 bucks each to spend . Mike used all his money to buy 3pens and 4books,while Stev bought 4pens and 3books and had a balance of 50 bucks.Find the cost of a pen and book.

$${Mike}\:{and}\:{Stev}\:{had}\:\mathrm{620}\:{bucks} \\ $$$${each}\:{to}\:{spend}\:.\:{Mike}\:{used}\:{all} \\ $$$${his}\:{money}\:{to}\:{buy}\:\mathrm{3}{pens}\:{and}\: \\ $$$$\mathrm{4}{books},{while}\:{Stev}\:{bought}\: \\ $$$$\mathrm{4}{pens}\:{and}\:\mathrm{3}{books}\:{and}\:{had}\:{a} \\ $$$${balance}\:{of}\:\mathrm{50}\:{bucks}.{Find} \\ $$$${the}\:{cost}\:{of}\:{a}\:{pen}\:{and}\:{book}. \\ $$$$ \\ $$

Question Number 40055 Answers: 1 Comments: 0

A boy starts from a point A and moves on a bearing of 20° to a point B which is 5km from A.He then changes his course to a bearing of 11° and moves to a point C which is 12km from B. Find the distance and bearing from C to A.

$${A}\:{boy}\:{starts}\:{from}\:{a}\:{point}\:{A} \\ $$$${and}\:{moves}\:{on}\:{a}\:{bearing}\:{of} \\ $$$$\mathrm{20}°\:{to}\:{a}\:{point}\:{B}\:{which}\:{is} \\ $$$$\mathrm{5}{km}\:{from}\:{A}.{He}\:{then}\:{changes} \\ $$$${his}\:{course}\:{to}\:{a}\:{bearing}\:{of}\: \\ $$$$\mathrm{11}°\:{and}\:{moves}\:{to}\:{a}\:{point}\:{C}\:{which}\:{is} \\ $$$$\mathrm{12}{km}\:{from}\:{B}. \\ $$$${Find}\:{the}\:{distance}\:{and}\:{bearing} \\ $$$${from}\:{C}\:{to}\:{A}. \\ $$$$ \\ $$

Question Number 40054 Answers: 0 Comments: 0

3 boys X,Y,and Z are standing 3 metres north of each other if X and Z are both 1.5m tall and Y is 2m tall. Find a) the bearing from each of the boys b) the bearing if Y moves to the left.

$$\mathrm{3}\:{boys}\:{X},{Y},{and}\:{Z}\:{are}\:{standing} \\ $$$$\mathrm{3}\:{metres}\:{north}\:{of}\:{each}\:{other} \\ $$$${if}\:{X}\:{and}\:{Z}\:{are}\:{both}\:\mathrm{1}.\mathrm{5}{m}\:{tall} \\ $$$${and}\:{Y}\:{is}\:\mathrm{2}{m}\:{tall}. \\ $$$${Find} \\ $$$$\left.{a}\right)\:{the}\:{bearing}\:{from}\:{each}\:{of} \\ $$$${the}\:{boys} \\ $$$$\left.{b}\right)\:{the}\:{bearing}\:{if}\:{Y}\:{moves} \\ $$$${to}\:{the}\:{left}. \\ $$

Question Number 40140 Answers: 1 Comments: 1

calculate ∫_0 ^(π/2) ((sin(x)dx)/(cos^2 x +a^2 sin^2 x))dx

$${calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{sin}\left({x}\right){dx}}{{cos}^{\mathrm{2}} {x}\:+{a}^{\mathrm{2}} \:{sin}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 40052 Answers: 2 Comments: 0

f ′(x)=g(x) and g ′(x)=−f(x) for all real x andf(5)=2=f ′(5) then f^2 (10)+g^2 (10) is (a) 2 (b) 4 (c) 8 (d) none

$${f}\:'\left({x}\right)={g}\left({x}\right)\:{and}\:\:{g}\:'\left({x}\right)=−{f}\left({x}\right)\:{for} \\ $$$${all}\:{real}\:\:{x}\:\:{andf}\left(\mathrm{5}\right)=\mathrm{2}={f}\:'\left(\mathrm{5}\right)\:{then} \\ $$$${f}^{\mathrm{2}} \left(\mathrm{10}\right)+{g}^{\mathrm{2}} \left(\mathrm{10}\right)\:{is} \\ $$$$\left({a}\right)\:\:\:\mathrm{2}\:\:\:\:\:\left({b}\right)\:\:\:\mathrm{4}\:\:\:\:\:\left({c}\right)\:\:\:\:\mathrm{8}\:\:\:\:\:\left({d}\right)\:{none} \\ $$

Question Number 40048 Answers: 0 Comments: 2

Question Number 40047 Answers: 0 Comments: 0

let S_n = Σ_(k=0) ^n (((−1)^k )/(2k+1)) 1) give S_n interms of H_n 2)find lim_(n→+∞) S_n 3) let W_n = Σ_(k=0) ^n (((−1)^k )/(4k^2 −1)) find W_n interms of H_n calculate lim_(n→+∞) W_n

$${let}\:{S}_{{n}} \:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{S}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} {S}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{W}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{4}{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$${find}\:{W}_{{n}} \:\:{interms}\:{of}\:{H}_{{n}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{W}_{{n}} \\ $$

Question Number 40046 Answers: 0 Comments: 2

let S_n = Σ_(k=2) ^n (((−1)^k )/(k^2 −1)) 1) calculate S_n interms of H_n ( H_n =Σ_(k=1) ^n (1/k)) 2) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{S}_{{n}} \:\:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left(\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$$$ \\ $$

Question Number 40044 Answers: 0 Comments: 2

let f(t) = ∫_0 ^(π/2) ln( cosx +t sinx) 1) calculate f(0) 2) calculate f^′ (t) then find a simple form of f(t) 3) calculate ∫_0 ^(π/2) ln(cosx +2 sinx)dx 4) calculate ∫_0 ^(π/2) ln((√3)cosx +sinx)dx

$${let}\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\:{cosx}\:+{t}\:{sinx}\right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}\left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({t}\right)\:{then}\:{find}\:\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right) \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({cosx}\:+\mathrm{2}\:{sinx}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left(\sqrt{\mathrm{3}}{cosx}\:+{sinx}\right){dx} \\ $$

Question Number 40043 Answers: 1 Comments: 0

find the value of ∫_(−1) ^(+∞) (√(x+1))e^(−x) dx

$${find}\:\:{the}\:{value}\:{of}\:\:\int_{−\mathrm{1}} ^{+\infty} \:\:\sqrt{{x}+\mathrm{1}}{e}^{−{x}} \:{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 40042 Answers: 0 Comments: 0

1) find the roots of p(x)=(1+ix +x^2 )^n −(1−ix+x^2 )^n with n integr natural 2) factorize p(x) inside C(x) 3) give p(x) at form Σ a_p x^p

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{roots}\:{of}\:\: \\ $$$${p}\left({x}\right)=\left(\mathrm{1}+{ix}\:+{x}^{\mathrm{2}} \right)^{{n}} −\left(\mathrm{1}−{ix}+{x}^{\mathrm{2}} \right)^{{n}} \:{with}\:{n}\:{integr} \\ $$$${natural} \\ $$$$\left.\mathrm{2}\right)\:{factorize}\:{p}\left({x}\right)\:{inside}\:\:{C}\left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{p}\left({x}\right)\:{at}\:{form}\:\:\Sigma\:{a}_{{p}} {x}^{{p}} \\ $$

Question Number 40040 Answers: 0 Comments: 1

let A_n = ∫_0 ^n e^(−n( x+2−[x])) dx with n integr natural 1) calculate A_n 2) find lim_(n→+∞) A_n 3) study the convergence of Σ_n A_n

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{e}^{−{n}\left(\:{x}+\mathrm{2}−\left[{x}\right]\right)} {dx}\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{study}\:{the}\:{convergence}\:{of}\:\:\:\sum_{{n}} {A}_{{n}} \\ $$

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