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Question Number 185013 Answers: 4 Comments: 3

Question Number 185012 Answers: 0 Comments: 2

Σ_(k=0) ^n sin(kx)=Σ_(k=0) ^n Im(e^(ikx) ) =Im(Σ_(k=0) ^n (e^(ix) )^k ) =Im(((1−e^(i(n+1)x) )/(1−e^(ix) ))) ((1−e^(i(n+1)x) )/(1−e^(ix) ))=((e^(i(n+1)(x/2)) (e^(−i(n+1)(x/2)) −e^(i(n+1)(x/2)) ))/(e^(i(x/2)) (e^(−i(x/2)) −e^(i(x/2)) ))) =e^(in(x/2)) ×((−2sin((n+1)(x/2)))/(−2sin((x/2)))) =e^(in(x/2)) ((sin((n+1)(x/2)))/(sin((x/2)))) Σ_(k=0) ^n sin(kx)=((sin((n+1)(x/2) ))/(sin((x/2))))Im(e^(in(x/2)) ) =((sin((n+1)(x/2)))/(sin((x/2))))sin(((nx)/2))

$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\left(\mathrm{kx}\right)=\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{Im}\left(\mathrm{e}^{\mathrm{ikx}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{Im}\left(\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\left(\mathrm{e}^{\mathrm{ix}} \right)^{\mathrm{k}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{Im}\left(\frac{\mathrm{1}−\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}} }{\mathrm{1}−\mathrm{e}^{\mathrm{ix}} }\right) \\ $$$$\frac{\mathrm{1}−\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\mathrm{x}} }{\mathrm{1}−\mathrm{e}^{\mathrm{ix}} }=\frac{\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}} \left(\mathrm{e}^{−\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}} −\mathrm{e}^{\mathrm{i}\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}} \right)}{\mathrm{e}^{\mathrm{i}\frac{\mathrm{x}}{\mathrm{2}}} \left(\mathrm{e}^{−\mathrm{i}\frac{\mathrm{x}}{\mathrm{2}}} −\mathrm{e}^{\mathrm{i}\frac{\mathrm{x}}{\mathrm{2}}} \right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{e}^{\mathrm{in}\frac{\mathrm{x}}{\mathrm{2}}} ×\frac{−\mathrm{2sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\right)}{−\mathrm{2sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{e}^{\mathrm{in}\frac{\mathrm{x}}{\mathrm{2}}} \frac{\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)} \\ $$$$\underset{\mathrm{k}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\left(\mathrm{kx}\right)=\frac{\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\:\right)}{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{Im}\left(\mathrm{e}^{\mathrm{in}\frac{\mathrm{x}}{\mathrm{2}}} \right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{sin}\left(\left(\mathrm{n}+\mathrm{1}\right)\frac{\mathrm{x}}{\mathrm{2}}\right)}{\mathrm{sin}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)}\mathrm{sin}\left(\frac{\mathrm{nx}}{\mathrm{2}}\right) \\ $$

Question Number 185011 Answers: 0 Comments: 0

Question Number 185010 Answers: 0 Comments: 0

Question Number 185009 Answers: 0 Comments: 0

Question Number 185008 Answers: 1 Comments: 0

Question Number 184994 Answers: 0 Comments: 1

(√(1+(√(2+(√(3+(√(4+....+(√(70))))))))))=?

$$\sqrt{\mathrm{1}+\sqrt{\mathrm{2}+\sqrt{\mathrm{3}+\sqrt{\mathrm{4}+....+\sqrt{\mathrm{70}}}}}}=? \\ $$

Question Number 184992 Answers: 1 Comments: 0

prove that: ∫_o ^1 ((sint)/(e^t −1))=Σ_(n=o) (1/(n^2 +1))

$${prove}\:{that}: \\ $$$$\int_{{o}} ^{\mathrm{1}} \frac{{sint}}{{e}^{{t}} −\mathrm{1}}=\underset{{n}={o}} {\sum}\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{1}} \\ $$

Question Number 185000 Answers: 0 Comments: 0

A man walks along straight path at a speed 4 ft/s. A spotlight is located on the ground 20 ft from the path and is kept focused on the man. At what rate is spotlight rotating when the man is 15 ft from the point on the path closest to the light?

$${A}\:{man}\:{walks}\:{along}\:{straight}\:{path}\: \\ $$$$\:{at}\:{a}\:{speed}\:\mathrm{4}\:{ft}/{s}.\:{A}\:{spotlight}\:{is} \\ $$$$\:{located}\:{on}\:{the}\:{ground}\:\mathrm{20}\:{ft}\:{from}\: \\ $$$$\:{the}\:{path}\:{and}\:{is}\:{kept}\:{focused}\:{on}\:{the}\:{man}. \\ $$$$\:{At}\:{what}\:{rate}\:{is}\:{spotlight}\:{rotating} \\ $$$$\:{when}\:{the}\:{man}\:{is}\:\mathrm{15}\:{ft}\:{from}\:{the}\: \\ $$$${point}\:{on}\:{the}\:{path}\:{closest}\:{to}\:{the}\:{light}?\: \\ $$$$\: \\ $$

Question Number 184989 Answers: 1 Comments: 0

2^x =4x x=? solution???

$$\mathrm{2}^{\mathrm{x}} =\mathrm{4x} \\ $$$$\mathrm{x}=? \\ $$$$\mathrm{solution}??? \\ $$

Question Number 184988 Answers: 1 Comments: 0

Question Number 185116 Answers: 1 Comments: 0

Question Number 184980 Answers: 2 Comments: 0

(√7) + (√6) = a (√7) − (√6) = ?

$$\sqrt{\mathrm{7}}\:\:+\:\:\sqrt{\mathrm{6}}\:\:=\:\:\mathrm{a} \\ $$$$\sqrt{\mathrm{7}}\:−\:\sqrt{\mathrm{6}}\:=\:? \\ $$

Question Number 184969 Answers: 1 Comments: 2

3 + (1/(6+(3^2 /(6+(5^2 /(6+(7^2 /(6+(9^2 /(6+......)) )) )) )) )) = 𝛑 provet that.

$$\:\: \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\mathrm{3}\:+\:\frac{\mathrm{1}}{\mathrm{6}+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{5}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{7}^{\mathrm{2}} }{\mathrm{6}+\frac{\mathrm{9}^{\mathrm{2}} }{\mathrm{6}+......}\:\:\:}\:\:}\:\:}\:\:\:\:}\:\:=\:\boldsymbol{\pi} \\ $$$$\:\:\:\:\boldsymbol{\mathrm{provet}}\:\:\boldsymbol{\mathrm{that}}. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 184960 Answers: 2 Comments: 0

Question Number 184959 Answers: 2 Comments: 0

Question Number 184958 Answers: 1 Comments: 0

Question Number 184955 Answers: 2 Comments: 0

Question Number 184944 Answers: 1 Comments: 0

please you help me Σ_(k=0) ^n sin(k)=??

$${please}\:{you}\:{help}\:{me} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{sin}\left({k}\right)=?? \\ $$

Question Number 184942 Answers: 0 Comments: 2

f(x )= x + ⌊ x + (( ⌊ x_ ^ ⌋)/(⌊ (x^( 2) /(1 +x^( 2) )) ⌋+1)) ⌋ ⇒ f^( −1) (x )=?

$$ \\ $$$$\:\:\:\:{f}\left({x}\:\right)=\:{x}\:+\:\lfloor\:\:{x}\:+\:\frac{\:\lfloor\:\:\underset{} {\overset{} {{x}}}\:\:\rfloor}{\lfloor\:\:\:\frac{{x}^{\:\mathrm{2}} }{\mathrm{1}\:+{x}^{\:\mathrm{2}} }\:\:\:\rfloor+\mathrm{1}}\:\rfloor \\ $$$$\:\:\:\:\:\:\Rightarrow\:\:{f}^{\:−\mathrm{1}} \:\left({x}\:\right)=? \\ $$

Question Number 184941 Answers: 0 Comments: 0

Determiner une relation entre les coeficients de x,y,z pour que z=x+y a_1 x+b_1 y+c_1 z=u a_2 x+b_2 y+c_2 z=v a_3 x+b_3 y+c_3 z=w

$${Determiner}\:{une}\:{relation}\:{entre} \\ $$$${les}\:{coeficients}\:{de}\:{x},{y},{z}\: \\ $$$${pour}\:{que}\:\:{z}={x}+{y} \\ $$$$ \\ $$$${a}_{\mathrm{1}} {x}+{b}_{\mathrm{1}} {y}+{c}_{\mathrm{1}} {z}={u} \\ $$$${a}_{\mathrm{2}} {x}+{b}_{\mathrm{2}} {y}+{c}_{\mathrm{2}} {z}={v} \\ $$$${a}_{\mathrm{3}} {x}+{b}_{\mathrm{3}} {y}+{c}_{\mathrm{3}} {z}={w} \\ $$$$ \\ $$

Question Number 184939 Answers: 1 Comments: 0

What′s the convergent equation of this series? x_1 ^2 +x_2 ^2 +x_3 ^2 +...+x_n ^2 Help!

$$\mathrm{What}'\mathrm{s}\:\mathrm{the}\:\mathrm{convergent}\:\mathrm{equation} \\ $$$$\mathrm{of}\:\mathrm{this}\:\mathrm{series}? \\ $$$$\mathrm{x}_{\mathrm{1}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{2}} ^{\mathrm{2}} +\mathrm{x}_{\mathrm{3}} ^{\mathrm{2}} +...+\mathrm{x}_{\mathrm{n}} ^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 184938 Answers: 2 Comments: 0

If , { (( f : [ (√2) , +∞ ) → R )),(( f (x ) = x^( 2) + ⌊ (( 1)/(1 − ⌊ x^( 2) ⌋)) ⌋ )) :} ⇒ ⌊ f^( −1) ( π ) ⌋ = ?

$$ \\ $$$$\:\:\:\:\:\mathrm{If}\:\:,\:\:\begin{cases}{\:\:{f}\:\::\:\:\left[\:\sqrt{\mathrm{2}}\:,\:+\infty\:\right)\:\rightarrow\:\mathbb{R}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}\\{\:\:\:{f}\:\left({x}\:\right)\:=\:{x}^{\:\mathrm{2}} \:\:+\:\lfloor\:\frac{\:\mathrm{1}}{\mathrm{1}\:−\:\lfloor\:{x}^{\:\mathrm{2}} \:\rfloor}\:\rfloor\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\:\:\lfloor\:\:{f}^{\:−\mathrm{1}} \:\left(\:\pi\:\right)\:\rfloor\:=\:? \\ $$$$ \\ $$$$ \\ $$

Question Number 184937 Answers: 0 Comments: 0

find the laplace transform of the differential equation below (dy/dt) + 5y(t) + 6∫_0 ^t y(τ)dτ = u(t) where y(0) = 2

$$\mathrm{find}\:\mathrm{the}\:\mathrm{laplace}\:\mathrm{transform}\:\mathrm{of}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation}\:\mathrm{below} \\ $$$$\frac{{dy}}{{dt}}\:+\:\mathrm{5}{y}\left({t}\right)\:+\:\mathrm{6}\int_{\mathrm{0}} ^{{t}} {y}\left(\tau\right){d}\tau\:=\:{u}\left({t}\right)\:\mathrm{where}\:{y}\left(\mathrm{0}\right)\:=\:\mathrm{2} \\ $$

Question Number 184936 Answers: 1 Comments: 0

Use Laplace transform to solve the differential equation ((d^2 v(t))/dt^2 ) +6((dv(t))/dt) + 8v(t) = 2u(t) when v(0) = 1 and v^• (0) = −2

$$\mathrm{Use}\:\mathrm{Laplace}\:\mathrm{transform}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{the}\:\mathrm{differential} \\ $$$$\mathrm{equation} \\ $$$$\:\frac{{d}^{\mathrm{2}} {v}\left({t}\right)}{{dt}^{\mathrm{2}} }\:+\mathrm{6}\frac{{dv}\left({t}\right)}{{dt}}\:+\:\mathrm{8}{v}\left({t}\right)\:=\:\mathrm{2}{u}\left({t}\right)\:\: \\ $$$$\mathrm{when}\:{v}\left(\mathrm{0}\right)\:=\:\mathrm{1}\:\mathrm{and}\:\overset{\bullet} {{v}}\left(\mathrm{0}\right)\:=\:−\mathrm{2} \\ $$

Question Number 184932 Answers: 1 Comments: 0

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