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Question Number 41519 Answers: 2 Comments: 0

let z=(√(2−(√3))) −i(√(2+(√3))) calculate ∣z^n ∣ and arg(z^n )

$${let}\:{z}=\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\:\:−{i}\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}} \\ $$$${calculate}\:\mid{z}^{{n}} \mid\:\:{and}\:{arg}\left({z}^{{n}} \right) \\ $$

Question Number 41518 Answers: 3 Comments: 0

calculate A_n = ∫_0 ^1 (1−t^2 )^n dt with n integr natural

$${calculate}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt}\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 41517 Answers: 1 Comments: 1

let S_n = 1 +(1/((^3 (√2)))) + (1/((^3 (√3)))) + ....+(1/((^3 (√n)))) calculate lim _(n→+∞) S_n

$${let}\:\:{S}_{{n}} =\:\mathrm{1}\:+\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{\mathrm{2}}\right)}\:+\:\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{\mathrm{3}}\right)}\:+\:....+\frac{\mathrm{1}}{\left(^{\mathrm{3}} \sqrt{{n}}\right)} \\ $$$${calculate}\:{lim}\:_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$

Question Number 41516 Answers: 1 Comments: 1

calculate ∫_0 ^1 ((ln(1+x))/((1+x)^4 )) dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{4}} }\:{dx} \\ $$

Question Number 41515 Answers: 1 Comments: 3

let f_n (x) =((sin(2(n+1)x))/(sinx)) if x∈]0,(π/2)] and f_n (0)=2(n+1) let u_n = ∫_0 ^(π/2) f_n (x)dx 1) prove that ∀n fromN u_(n+1) −u_n =2(((−1)^(n+1) )/(2n+3)) 2)find lim_(n→+∞) u_n

$$\left.{l}\left.{et}\:\:{f}_{{n}} \left({x}\right)\:=\frac{{sin}\left(\mathrm{2}\left({n}+\mathrm{1}\right){x}\right)}{{sinx}}\:{if}\:\:{x}\in\right]\mathrm{0},\frac{\pi}{\mathrm{2}}\right]\:{and}\:{f}_{{n}} \left(\mathrm{0}\right)=\mathrm{2}\left({n}+\mathrm{1}\right)\:\:{let} \\ $$$${u}_{{n}} =\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{f}_{{n}} \left({x}\right){dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\forall{n}\:{fromN}\:\:{u}_{{n}+\mathrm{1}} −{u}_{{n}} =\mathrm{2}\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\mathrm{2}{n}+\mathrm{3}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} \:{u}_{{n}} \\ $$

Question Number 41514 Answers: 1 Comments: 0

find ∫ cos(lnx)dx

$${find}\:\:\:\int\:\:{cos}\left({lnx}\right){dx}\: \\ $$

Question Number 41513 Answers: 2 Comments: 0

let A_n = ∫_0 ^∞ e^(−nx^2 ) cos(x^2 ) dx and B_n =∫_0 ^∞ e^(−nx^2 ) sin(x^2 )dx (n∈ N^★ ) 1) calculate A_n and B_n 2) find lim_(n→+∞) (A_n /B_n )

$${let}\:\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}^{\mathrm{2}} } {cos}\left({x}^{\mathrm{2}} \right)\:{dx}\:\:{and}\:\:{B}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{nx}^{\mathrm{2}} } {sin}\left({x}^{\mathrm{2}} \right){dx}\:\:\:\:\left({n}\in\:{N}^{\bigstar} \right) \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{A}_{{n}} \:{and}\:\:{B}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:\:\frac{{A}_{{n}} }{{B}_{{n}} } \\ $$

Question Number 41512 Answers: 0 Comments: 0

let u_n =(1/1^3 ) +(1/2^3 ) +....+(1/n^3 ) 1)prove that (9/8) −(1/(2(n+1)^2 )) ≤ u_n ≤ (3/2) −(1/(2n^2 )) 2) prove that ∀ n∈N^★ 1≤u_n ≤ (3/2) 3) prove that (u_n )is convegente .

$${let}\:\:{u}_{{n}} =\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} }\:+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }\:+....+\frac{\mathrm{1}}{{n}^{\mathrm{3}} } \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\:\:\frac{\mathrm{9}}{\mathrm{8}}\:−\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)^{\mathrm{2}} }\:\leqslant\:{u}_{{n}} \leqslant\:\frac{\mathrm{3}}{\mathrm{2}}\:−\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} } \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\forall\:{n}\in{N}^{\bigstar} \:\:\:\mathrm{1}\leqslant{u}_{{n}} \leqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\left({u}_{{n}} \right){is}\:{convegente}\:. \\ $$

Question Number 41508 Answers: 0 Comments: 0

A facial surgery is estimated to cost N25millon in 10 years time. A 50 years old woman that is expecting to go on retirement at 55years is planning to generate funds for the surgery at her retirement. ow much should she set aside for the surgery if the interest rate is 14.75% at the time of retirement? what is the present value of the cost of surgery?(as at the time the woman is 50years given the interest rate at that time is 11.5%)

Question Number 41495 Answers: 2 Comments: 3

The equation π^x =−2x^2 +6x−9 has

$$\mathrm{The}\:\mathrm{equation}\:\pi^{{x}} =−\mathrm{2}{x}^{\mathrm{2}} +\mathrm{6}{x}−\mathrm{9}\:\mathrm{has} \\ $$

Question Number 41488 Answers: 2 Comments: 0

Question Number 41487 Answers: 3 Comments: 4

Question Number 41461 Answers: 1 Comments: 3

Find area of square inserted in curve f(x)= 3x−x^3 .

$$\mathrm{Find}\:\mathrm{area}\:\mathrm{of}\:\mathrm{square}\:\mathrm{inserted}\:\mathrm{in}\:\mathrm{curve} \\ $$$$\mathrm{f}\left({x}\right)=\:\mathrm{3}{x}−{x}^{\mathrm{3}} . \\ $$

Question Number 41459 Answers: 0 Comments: 1

Question Number 41454 Answers: 1 Comments: 1

Evaluate Σ_(k=1) ^(2n−1) (−1)^(k−1) k^3

$$\mathrm{Evaluate}\:\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}{n}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} {k}^{\mathrm{3}} \\ $$

Question Number 41447 Answers: 1 Comments: 0

Question Number 41444 Answers: 0 Comments: 1

Given that cos(θ + (π/3))= (1/2) find the values of θ inthe range 0°≤ θ≤360°

$${Given}\:{that}\:\:\:{cos}\left(\theta\:+\:\frac{\pi}{\mathrm{3}}\right)=\:\frac{\mathrm{1}}{\mathrm{2}}\:{find}\:{the}\:{values}\:{of}\:\theta\:{inthe}\:{range}\:\:\mathrm{0}°\leqslant\:\theta\leqslant\mathrm{360}° \\ $$

Question Number 41443 Answers: 1 Comments: 0

Suppose that you wish to fabricate a uniform wire out of 1 gram of copper.If the wire is to have a resistance of R=0.5Ω,and all of the copper is to be used.What will be(i)the length and (ii)the diameter of the wire.

$${Suppose}\:{that}\:{you}\:{wish}\:{to}\:{fabricate} \\ $$$${a}\:{uniform}\:{wire}\:{out}\:{of}\:\mathrm{1}\:{gram}\:{of} \\ $$$${copper}.{If}\:{the}\:{wire}\:{is}\:{to}\:{have}\:{a} \\ $$$${resistance}\:{of}\:{R}=\mathrm{0}.\mathrm{5}\Omega,{and}\:{all}\:{of} \\ $$$${the}\:{copper}\:{is}\:{to}\:{be}\:{used}.{What}\:{will} \\ $$$${be}\left({i}\right){the}\:{length}\:{and}\:\left({ii}\right){the}\:{diameter} \\ $$$${of}\:{the}\:{wire}. \\ $$

Question Number 41479 Answers: 1 Comments: 0