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Question Number 55724    Answers: 0   Comments: 4

Question Number 55707    Answers: 0   Comments: 0

Question Number 55706    Answers: 0   Comments: 0

Question Number 55705    Answers: 0   Comments: 0

Question Number 55704    Answers: 1   Comments: 0

Prove that: 2 sin (1/2)θcos (3/2)θ+2sin (5/2)θ +2 sin (3/2)θ+2sin (3/2)θcos (7/2)θ =sin 4θ+sin 5θ

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{2}\:\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}\theta\mathrm{cos}\:\frac{\mathrm{3}}{\mathrm{2}}\theta+\mathrm{2sin}\:\frac{\mathrm{5}}{\mathrm{2}}\theta\: \\ $$$$+\mathrm{2}\:\mathrm{sin}\:\frac{\mathrm{3}}{\mathrm{2}}\theta+\mathrm{2sin}\:\frac{\mathrm{3}}{\mathrm{2}}\theta\mathrm{cos}\:\frac{\mathrm{7}}{\mathrm{2}}\theta \\ $$$$=\mathrm{sin}\:\mathrm{4}\theta+\mathrm{sin}\:\mathrm{5}\theta \\ $$

Question Number 55702    Answers: 0   Comments: 5

s=∫_0 ^( x) (√(1+(3t^2 +p)^2 ))dt = ? take p=1 for a special case.

$${s}=\int_{\mathrm{0}} ^{\:{x}} \sqrt{\mathrm{1}+\left(\mathrm{3}{t}^{\mathrm{2}} +{p}\right)^{\mathrm{2}} }{dt}\:\:=\:? \\ $$$$\:\:\:\:{take}\:{p}=\mathrm{1}\:\:{for}\:{a}\:{special}\:{case}. \\ $$

Question Number 55701    Answers: 1   Comments: 1

Is true or not that 4181 is the only one Fibonacci′s number with no prime factor which is also a Fibonacci′s number?

$${Is}\:{true}\:{or}\:{not}\:{that}\:\mathrm{4181}\:{is}\:{the}\:{only}\:{one} \\ $$$${Fibonacci}'{s}\:{number}\:{with}\:{no}\:{prime}\:{factor} \\ $$$${which}\:{is}\:{also}\:{a}\:{Fibonacci}'{s}\:{number}? \\ $$

Question Number 55685    Answers: 1   Comments: 0

proof that Σ_(i=1) ^n (a_i /(a_i −x))=2015 has exactly n real roots.o<a_1 ....<a_n

$${proof}\:{that}\: \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{a}_{{i}} }{{a}_{{i}} −{x}}=\mathrm{2015}\:{has}\:{exactly}\:{n}\:{real}\: \\ $$$${roots}.{o}<{a}_{\mathrm{1}} ....<{a}_{{n}} \\ $$

Question Number 55675    Answers: 0   Comments: 0

Question Number 55674    Answers: 1   Comments: 0

The smallest integer numbers with n ≥ 2018 so ((√3)+3i)^n form real numbers is..

$$\mathrm{The}\:\mathrm{smallest}\:\mathrm{integer}\:\mathrm{numbers} \\ $$$$\mathrm{with}\:{n}\:\geqslant\:\mathrm{2018}\:\mathrm{so} \\ $$$$\left(\sqrt{\mathrm{3}}+\mathrm{3}{i}\right)^{{n}} \:\mathrm{form}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{is}.. \\ $$

Question Number 55673    Answers: 0   Comments: 0

f(z)=z Re(z)+z^ Im(z) +z^ f′(z_0 )=...

$${f}\left({z}\right)={z}\:\mathrm{Re}\left({z}\right)+\bar {{z}}\:\mathrm{Im}\left({z}\right)\:+\bar {{z}}\: \\ $$$${f}'\left({z}_{\mathrm{0}} \right)=... \\ $$

Question Number 55672    Answers: 0   Comments: 0

The value of complex integral ∫_(∣z∣=1) (z^2 sin (1/z)+(1/z^2 )sin z) dz is...

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{complex}\:\mathrm{integral} \\ $$$$\int_{\mid{z}\mid=\mathrm{1}} \left({z}^{\mathrm{2}} \mathrm{sin}\:\frac{\mathrm{1}}{{z}}+\frac{\mathrm{1}}{{z}^{\mathrm{2}} }\mathrm{sin}\:{z}\right)\:{dz}\:\mathrm{is}... \\ $$

Question Number 55671    Answers: 0   Comments: 0

Let z ∈ C , so ∣1+z^2 ∣<1. Prove that 2∣1+z^2 ∣≥1

$$\mathrm{Let}\:\mathrm{z}\:\in\:\mathbb{C}\:,\:\mathrm{so}\:\mid\mathrm{1}+{z}^{\mathrm{2}} \mid<\mathrm{1}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{2}\mid\mathrm{1}+{z}^{\mathrm{2}} \mid\geqslant\mathrm{1} \\ $$

Question Number 55669    Answers: 0   Comments: 0

Question Number 55668    Answers: 1   Comments: 3

Find all functions y=f(x) such that y′y′′=y′′′.

$${Find}\:{all}\:{functions}\:{y}={f}\left({x}\right)\:{such}\:{that} \\ $$$${y}'{y}''={y}'''. \\ $$

Question Number 55666    Answers: 0   Comments: 0

Question Number 55665    Answers: 0   Comments: 0

Question Number 55663    Answers: 2   Comments: 0

Question Number 55658    Answers: 1   Comments: 0

find the difference of the roots of the following quadratic equation (3+2(√(2 )))x^2 +(1+(√2))x =2

$$\mathrm{find}\:\mathrm{the}\:\mathrm{difference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\: \\ $$$$\mathrm{following}\:\mathrm{quadratic}\:\mathrm{equation} \\ $$$$\left(\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}\:}\right)\mathrm{x}^{\mathrm{2}} \:+\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)\mathrm{x}\:=\mathrm{2} \\ $$

Question Number 55643    Answers: 0   Comments: 1

known function f diferensiable continues at [a, b] If f(a)=f(b)=0 and ∫_a ^b [f(x)]^2 dx=1 Prove that ∫_a ^b x^2 [f′(x)]^2 dx ≥(1/4)

$$\mathrm{known}\:\mathrm{function}\:{f} \\ $$$$\mathrm{diferensiable}\:\mathrm{continues}\:\mathrm{at}\:\left[{a},\:{b}\right] \\ $$$$\mathrm{If}\:{f}\left({a}\right)={f}\left({b}\right)=\mathrm{0} \\ $$$$\mathrm{and}\: \\ $$$$\int_{{a}} ^{{b}} \left[{f}\left({x}\right)\right]^{\mathrm{2}} {dx}=\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\int_{{a}} ^{{b}} {x}^{\mathrm{2}} \left[{f}'\left({x}\right)\right]^{\mathrm{2}} \:{dx}\:\geqslant\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Question Number 55642    Answers: 0   Comments: 0

Prove the following statements: If for every n , f_n form ascend function and {f_n } uniform convergences to f at [a, b], then lim_(n→∞) ∫_a ^b f_n (x) dx →∫_a ^b f(x) dx

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statements}: \\ $$$$\mathrm{If}\:\mathrm{for}\:\mathrm{every}\:{n}\:,\:{f}_{{n}} \:\mathrm{form}\:\mathrm{ascend}\:\mathrm{function} \\ $$$$\mathrm{and}\:\left\{{f}_{{n}} \right\}\:\mathrm{uniform}\:\mathrm{convergences} \\ $$$$\mathrm{to}\:{f}\:\mathrm{at}\:\left[{a},\:{b}\right],\:\mathrm{then} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{{a}} ^{{b}} {f}_{{n}} \left({x}\right)\:{dx}\:\rightarrow\int_{{a}} ^{{b}} {f}\left({x}\right)\:{dx} \\ $$

Question Number 55641    Answers: 0   Comments: 1

Studies of convergences the numbers real sequence {x_n }, with x_1 =1 and x_(n+1) =((x_n ^2 +2)/(2x_n )), n≥1

$$\mathrm{Studies}\:\mathrm{of}\:\mathrm{convergences} \\ $$$$\mathrm{the}\:\mathrm{numbers}\:\mathrm{real}\:\mathrm{sequence}\:\left\{{x}_{{n}} \right\}, \\ $$$$\mathrm{with}\:{x}_{\mathrm{1}} =\mathrm{1}\:\mathrm{and}\:{x}_{{n}+\mathrm{1}} =\frac{{x}_{{n}} ^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}_{{n}} },\:{n}\geqslant\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

Question Number 55640    Answers: 0   Comments: 0

known real numbers sequence {a_n } and {b_n } both of them convergences to 0. If {b_n } monotonous descend and lim_(n→∞) ((a_(n+1) −a_n )/(b_(n+1) −b_n )) . then lim_(n→∞) (a_n /(2b_n ))=..

$$\mathrm{known}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{sequence} \\ $$$$\left\{{a}_{{n}} \right\}\:\mathrm{and}\:\left\{{b}_{{n}} \right\}\:\mathrm{both}\:\mathrm{of}\:\mathrm{them} \\ $$$$\mathrm{convergences}\:\mathrm{to}\:\mathrm{0}. \\ $$$$\mathrm{If}\:\left\{{b}_{{n}} \right\}\:\mathrm{monotonous}\:\mathrm{descend} \\ $$$$\mathrm{and}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{a}_{{n}+\mathrm{1}} −{a}_{{n}} }{{b}_{{n}+\mathrm{1}} −{b}_{{n}} }\:. \\ $$$$\mathrm{then}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{a}_{{n}} }{\mathrm{2}{b}_{{n}} }=.. \\ $$

Question Number 55639    Answers: 1   Comments: 0

Known a ∈ R and function f : R→R satiesfied ∣xf(x)+a∣ < sin^2 (x−a). For all x ∈ R value of lim_(x→a) f(x) ..

$$\mathrm{Known}\:{a}\:\in\:\mathbb{R}\:\mathrm{and} \\ $$$$\mathrm{function}\:{f}\::\:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{satiesfied} \\ $$$$\mid{xf}\left({x}\right)+{a}\mid\:<\:\mathrm{sin}^{\mathrm{2}} \:\left({x}−{a}\right).\: \\ $$$$\mathrm{For}\:\mathrm{all}\:{x}\:\in\:\mathbb{R} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\:{f}\left({x}\right)\:.. \\ $$

Question Number 55638    Answers: 1   Comments: 0

For all n ∈ N f_n (x)= { ((((nx)/(2n−1)), x ∈ [0, ((2n−1)/n)])),((1 , x ∈[((2n−1)/n), 2])) :} then for n→∞ ∫_1 ^2 f_n (x) dx convergences to..

$$\mathrm{For}\:\mathrm{all}\:{n}\:\in\:\mathbb{N} \\ $$$${f}_{{n}} \left({x}\right)=\begin{cases}{\frac{{nx}}{\mathrm{2}{n}−\mathrm{1}},\:\:\:\:\:{x}\:\in\:\left[\mathrm{0},\:\frac{\mathrm{2}{n}−\mathrm{1}}{{n}}\right]}\\{\mathrm{1}\:\:\:\:\:\:\:\:\:\:,\:\:\:\:\:\:{x}\:\in\left[\frac{\mathrm{2}{n}−\mathrm{1}}{{n}},\:\mathrm{2}\right]}\end{cases} \\ $$$$\mathrm{then}\:\mathrm{for}\:{n}\rightarrow\infty \\ $$$$\int_{\mathrm{1}} ^{\mathrm{2}} {f}_{{n}} \left({x}\right)\:{dx}\:\mathrm{convergences}\:\mathrm{to}.. \\ $$$$ \\ $$

Question Number 55637    Answers: 2   Comments: 0

Value of lim_(n→∞) n ∫_0 ^1 ((2x^n )/(x+x^(2n+1) )) dx=..

$$\mathrm{Value}\:\mathrm{of}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{n}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{2}{x}^{{n}} }{{x}+{x}^{\mathrm{2}{n}+\mathrm{1}} }\:{dx}=.. \\ $$

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