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Question Number 213607 Answers: 0 Comments: 0

Give a,b,c > 0 such that ab+bc+ca=abc Prove that : a+b+c≥4((a/(bc))+(b/(ca))+(c/(ab)))+5

$${Give}\:{a},{b},{c}\:>\:\mathrm{0}\:{such}\:{that} \\ $$$${ab}+{bc}+{ca}={abc} \\ $$$${Prove}\:{that}\:: \\ $$$${a}+{b}+{c}\geqslant\mathrm{4}\left(\frac{{a}}{{bc}}+\frac{{b}}{{ca}}+\frac{{c}}{{ab}}\right)+\mathrm{5} \\ $$

Question Number 213606 Answers: 2 Comments: 0

△ABC. 2a+b=2c. Find the minimum of (3/(sin C)) + (1/(tan A)).

$$\bigtriangleup{ABC}.\:\mathrm{2}{a}+{b}=\mathrm{2}{c}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{of} \\ $$$$\frac{\mathrm{3}}{\mathrm{sin}\:{C}}\:+\:\frac{\mathrm{1}}{\mathrm{tan}\:{A}}. \\ $$

Question Number 213604 Answers: 0 Comments: 0

show that ∫_C e^z^3 dz=0 where C is any simple closed contour. Evaluate the integral ∫_( C_1 ) f(z)dz , ∫_( C_2 ) f(z)dz where f(z)=(y−x)−3x^2 i C_3 =OA ; z(y)=x+iy=iy , (0≤y≤1) C_1 =AB ; z(x)=x+iy=x+i , (0≤x≤1) C_2 =OB ; z(x)=x+iy=x+ix , (0≤x≤1) Let′s C be the quadrant z=2e^(iθ) ,0≤θ≤(π/2) show that ∣∫_( C) ((z^ +4)/(z^3 −1)) dz∣≤((6π)/7) Let C be any simple closed contour described in the positive sense in the z plane and write g(z)=∫_( C) ((s^3 +2s)/((z−2s)^3 )) ds show that g(z)=6πiz when z is inside C show that g(z)=0 when z is outside C

$$\mathrm{show}\:\mathrm{that}\:\:\int_{\boldsymbol{\mathcal{C}}} \:{e}^{{z}^{\mathrm{3}} } \:\mathrm{d}{z}=\mathrm{0} \\ $$$$\mathrm{where}\:\boldsymbol{\mathcal{C}}\:\mathrm{is}\:\mathrm{any}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{contour}. \\ $$$$\:\:\:\: \\ $$$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int_{\:{C}_{\mathrm{1}} } {f}\left({z}\right)\mathrm{d}{z}\:,\:\int_{\:{C}_{\mathrm{2}} } {f}\left({z}\right)\mathrm{d}{z} \\ $$$$\mathrm{where}\:{f}\left({z}\right)=\left({y}−{x}\right)−\mathrm{3}{x}^{\mathrm{2}} \boldsymbol{{i}} \\ $$$${C}_{\mathrm{3}} ={OA}\:;\:{z}\left({y}\right)={x}+\boldsymbol{{i}}{y}=\boldsymbol{{i}}{y}\:,\:\left(\mathrm{0}\leq{y}\leq\mathrm{1}\right) \\ $$$${C}_{\mathrm{1}} ={AB}\:;\:{z}\left({x}\right)={x}+\boldsymbol{{i}}{y}={x}+\boldsymbol{{i}}\:,\:\left(\mathrm{0}\leq{x}\leq\mathrm{1}\right) \\ $$$${C}_{\mathrm{2}} ={OB}\:;\:{z}\left({x}\right)={x}+\boldsymbol{{i}}{y}={x}+\boldsymbol{{i}}{x}\:,\:\left(\mathrm{0}\leq{x}\leq\mathrm{1}\right) \\ $$$$\: \\ $$$$\mathrm{Let}'\mathrm{s}\:{C}\:\mathrm{be}\:\mathrm{the}\:\mathrm{quadrant} \\ $$$${z}=\mathrm{2}{e}^{\boldsymbol{{i}}\theta} \:,\mathrm{0}\leq\theta\leq\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\mid\int_{\:{C}} \:\frac{{z}^{\:} +\mathrm{4}}{{z}^{\mathrm{3}} −\mathrm{1}}\:\mathrm{d}{z}\mid\leq\frac{\mathrm{6}\pi}{\mathrm{7}} \\ $$$$\: \\ $$$$\mathrm{Let}\:{C}\:\mathrm{be}\:\mathrm{any}\:\mathrm{simple}\:\mathrm{closed}\:\mathrm{contour} \\ $$$$\mathrm{described}\:\mathrm{in}\:\mathrm{the}\:\mathrm{positive}\:\mathrm{sense}\:\mathrm{in}\:\mathrm{the}\:{z}\:\mathrm{plane} \\ $$$$\mathrm{and}\:\mathrm{write} \\ $$$$\mathrm{g}\left({z}\right)=\int_{\:{C}} \:\:\frac{{s}^{\mathrm{3}} +\mathrm{2}{s}}{\left({z}−\mathrm{2}{s}\right)^{\mathrm{3}} }\:\mathrm{d}{s} \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{g}\left({z}\right)=\mathrm{6}\pi\boldsymbol{{i}}{z}\:\mathrm{when}\:{z}\:\mathrm{is}\:\mathrm{inside}\:{C}\: \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{g}\left({z}\right)=\mathrm{0}\:\mathrm{when}\:{z}\:\mathrm{is}\:\mathrm{outside}\:{C} \\ $$

Question Number 213589 Answers: 2 Comments: 0

Find: (((3 - (3/4))∙(3 - (3/5))∙(3 - (1/2))∙(3 - (3/7))∙...∙(3 - (1/6)))/(27^5 )) = ?

$$\mathrm{Find}: \\ $$$$\frac{\left(\mathrm{3}\:-\:\frac{\mathrm{3}}{\mathrm{4}}\right)\centerdot\left(\mathrm{3}\:-\:\frac{\mathrm{3}}{\mathrm{5}}\right)\centerdot\left(\mathrm{3}\:-\:\frac{\mathrm{1}}{\mathrm{2}}\right)\centerdot\left(\mathrm{3}\:-\:\frac{\mathrm{3}}{\mathrm{7}}\right)\centerdot...\centerdot\left(\mathrm{3}\:-\:\frac{\mathrm{1}}{\mathrm{6}}\right)}{\mathrm{27}^{\mathrm{5}} }\:=\:? \\ $$

Question Number 213580 Answers: 0 Comments: 2

Question Number 213579 Answers: 0 Comments: 1

Question Number 213575 Answers: 0 Comments: 0

Question Number 213573 Answers: 1 Comments: 3

Question Number 213556 Answers: 3 Comments: 1

Question Number 213555 Answers: 2 Comments: 0

f(z)=Σ_(j=−∞) ^∞ (z/(z^2 +j^2 )) , z∈(0,∞) lim_(z→∞) f(z)=??

$${f}\left({z}\right)=\underset{{j}=−\infty} {\overset{\infty} {\sum}}\:\frac{{z}}{{z}^{\mathrm{2}} +{j}^{\mathrm{2}} }\:,\:{z}\in\left(\mathrm{0},\infty\right) \\ $$$$\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({z}\right)=?? \\ $$

Question Number 213550 Answers: 2 Comments: 1

Question Number 213548 Answers: 1 Comments: 0

0<c<1 such that the recursive sequence {a_n } defined by setting a_(1 ) = (c/2) , a_(n+1) = (1/2)(c+a_n ^2 ) for n∈ N monotonic and convergent

$$\mathrm{0}<{c}<\mathrm{1}\:\mathrm{such}\:\mathrm{that}\:\mathrm{the}\:\mathrm{recursive}\:\mathrm{sequence} \\ $$$$\left\{{a}_{{n}} \right\}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{setting}\: \\ $$$$\:\mathrm{a}_{\mathrm{1}\:} =\:\frac{\mathrm{c}}{\mathrm{2}}\:\:,\:{a}_{\mathrm{n}+\mathrm{1}} \:=\:\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{c}+\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \right)\:\:\mathrm{for}\:\mathrm{n}\in\:\mathbb{N} \\ $$$$\mathrm{monotonic}\:\mathrm{and}\:\mathrm{convergent} \\ $$

Question Number 213530 Answers: 1 Comments: 1

Question Number 213534 Answers: 1 Comments: 1

Question Number 213522 Answers: 0 Comments: 0

The two corner points of a square lie on curve f(x)= x^2 −2x−3 and the other two corner points lie on curve g(x)= −x^2 +2x+3 . It is known that the area of a square can be expressed by p+q(√r) , for a natural number p,q ,r where r is not divisible by any perfect square number other 1. The value of p+q+r =?

$$\:\:\mathrm{The}\:\mathrm{two}\:\mathrm{corner}\:\mathrm{points}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\: \\ $$$$\:\:\mathrm{lie}\:\mathrm{on}\:\mathrm{curve}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}\:\mathrm{and}\: \\ $$$$\:\mathrm{the}\:\mathrm{other}\:\mathrm{two}\:\mathrm{corner}\:\mathrm{points}\:\mathrm{lie}\:\mathrm{on}\: \\ $$$$\:\mathrm{curve}\:\mathrm{g}\left(\mathrm{x}\right)=\:−\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{3}\:.\:\mathrm{It}\:\mathrm{is}\:\mathrm{known} \\ $$$$\:\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{can}\:\mathrm{be}\: \\ $$$$\:\:\mathrm{expressed}\:\mathrm{by}\:\mathrm{p}+\mathrm{q}\sqrt{\mathrm{r}}\:,\:\mathrm{for}\:\mathrm{a}\:\mathrm{natural}\: \\ $$$$\:\mathrm{number}\:\mathrm{p},\mathrm{q}\:,\mathrm{r}\:\mathrm{where}\:\mathrm{r}\:\mathrm{is}\:\mathrm{not}\:\mathrm{divisible}\: \\ $$$$\:\mathrm{by}\:\mathrm{any}\:\mathrm{perfect}\:\mathrm{square}\:\mathrm{number}\:\mathrm{other}\:\mathrm{1}. \\ $$$$\:\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\mathrm{p}+\mathrm{q}+\mathrm{r}\:=? \\ $$

Question Number 213518 Answers: 1 Comments: 0

∫_0 ^( 2π) ((z∙sin(z))/(1+cos^2 (z))) dz ∫_( ∣z∣=2) (1/(z^2 +1)) dz ∫_( ∣z∣=2) ((sin(z))/(z^2 +1)) dz

$$\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \:\frac{{z}\centerdot\mathrm{sin}\left({z}\right)}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \left({z}\right)}\:\mathrm{d}{z} \\ $$$$\int_{\:\mid{z}\mid=\mathrm{2}} \:\frac{\mathrm{1}}{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{z} \\ $$$$\int_{\:\mid{z}\mid=\mathrm{2}} \:\frac{\mathrm{sin}\left({z}\right)}{{z}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{z} \\ $$

Question Number 213511 Answers: 4 Comments: 0

lim_(n→∞) [Σ_(r=1) ^n (1/2^r )] where [•] greatest integer finction

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\:\left[\underset{\mathrm{r}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{r}} }\right] \\ $$$$\:\:\:\mathrm{where}\:\left[\bullet\right]\:\mathrm{greatest}\:\mathrm{integer}\:\mathrm{finction} \\ $$

Question Number 213564 Answers: 1 Comments: 6

f: Z→R such that f(x).f(y)=f(x+y)+f(x−y) ⇒f(x)=¿

$${f}:\:{Z}\rightarrow{R}\:{such}\:{that} \\ $$$${f}\left({x}\right).{f}\left({y}\right)={f}\left({x}+{y}\right)+{f}\left({x}−{y}\right) \\ $$$$\Rightarrow{f}\left({x}\right)=¿ \\ $$

Question Number 213504 Answers: 1 Comments: 0