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Question Number 191369 Answers: 1 Comments: 0

calculate 𝛗= Σ_(n=1) ^∞ (( sin(((nπ)/3) ))/((2n + 1 )^( 2) ))= ?

$$ \\ $$$$\:\:\:{calculate} \\ $$$$ \\ $$$$\:\:\:\:\:\boldsymbol{\phi}=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\:{sin}\left(\frac{{n}\pi}{\mathrm{3}}\:\right)}{\left(\mathrm{2}{n}\:+\:\mathrm{1}\:\right)^{\:\mathrm{2}} }=\:? \\ $$$$ \\ $$

Question Number 191364 Answers: 2 Comments: 2

Question Number 191359 Answers: 0 Comments: 0

Find lim_(x→0) (𝚪(x)𝛙^((0)) (x)+π^2 ((cot(πx))/(sin(πx))))=?

$$\boldsymbol{\mathrm{Find}}\:\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\boldsymbol{\Gamma}\left(\boldsymbol{\mathrm{x}}\right)\boldsymbol{\psi}^{\left(\mathrm{0}\right)} \left(\boldsymbol{\mathrm{x}}\right)+\pi^{\mathrm{2}} \frac{\boldsymbol{\mathrm{cot}}\left(\pi\boldsymbol{\mathrm{x}}\right)}{\boldsymbol{\mathrm{sin}}\left(\pi\boldsymbol{\mathrm{x}}\right)}\right)=? \\ $$$$ \\ $$

Question Number 191344 Answers: 3 Comments: 0

if sin(x) + (√3) cos (x) = (1/2) ⇒(√3) sin(4x ) − cos (4x )= ?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{if}\:\:\:{sin}\left({x}\right)\:+\:\sqrt{\mathrm{3}}\:{cos}\:\left({x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\Rightarrow\sqrt{\mathrm{3}}\:\:{sin}\left(\mathrm{4}{x}\:\right)\:−\:\:{cos}\:\left(\mathrm{4}{x}\:\right)=\:? \\ $$$$ \\ $$

Question Number 191343 Answers: 1 Comments: 0

solve in R ⌊ (1/x) ⌋ + ⌊ (2/x) ⌋ + ⌊ (3/x) ⌋ = 1

$$ \\ $$$$\:\:\:\:\:\:\mathrm{solve}\:\:{in}\:\:\mathbb{R} \\ $$$$ \\ $$$$\:\:\:\:\:\:\lfloor\:\frac{\mathrm{1}}{{x}}\:\rfloor\:+\:\lfloor\:\frac{\mathrm{2}}{{x}}\:\rfloor\:+\:\lfloor\:\frac{\mathrm{3}}{{x}}\:\rfloor\:=\:\mathrm{1} \\ $$$$ \\ $$

Question Number 191342 Answers: 1 Comments: 0

calculate... Ω ={ ∫_0 ^( (π/4)) ( 1 + (√3) sin(x) + cos(x) )^( n) dx}^(1/n) = ?

$$ \\ $$$$\:\:\:\:{calculate}... \\ $$$$\:\Omega\:=\left\{\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left(\:\:\mathrm{1}\:+\:\sqrt{\mathrm{3}}\:{sin}\left({x}\right)\:+\:{cos}\left({x}\right)\:\right)^{\:{n}} {dx}\right\}^{\frac{\mathrm{1}}{{n}}} =\:? \\ $$$$ \\ $$$$ \\ $$

Question Number 191310 Answers: 1 Comments: 0

Question Number 191307 Answers: 1 Comments: 0

Question Number 191305 Answers: 1 Comments: 1

Question Number 191303 Answers: 2 Comments: 0

Question Number 191302 Answers: 1 Comments: 0

Question Number 191301 Answers: 2 Comments: 0

Question Number 191300 Answers: 1 Comments: 0

Question Number 191304 Answers: 2 Comments: 0

Question Number 191278 Answers: 1 Comments: 0

Question Number 191277 Answers: 0 Comments: 0

Question Number 191276 Answers: 1 Comments: 0

Question Number 191275 Answers: 1 Comments: 0

Question Number 191274 Answers: 1 Comments: 2

Question Number 191335 Answers: 1 Comments: 0

Question Number 191325 Answers: 2 Comments: 0

Question Number 191324 Answers: 1 Comments: 0

Question Number 191323 Answers: 1 Comments: 0

Question Number 191254 Answers: 0 Comments: 6

a=((x−a)/(x−b)) solve for x did i make anything wrong in the following? starpoint: ln∣a∣=ln∣((x−a)/(x−b))∣ ln∣a∣=ln∣x−a∣−ln∣x−b∣ ln∣a∣=ln∣(x/a)∣−ln∣(x/b)∣ ln∣a∣=ln∣x∣−ln∣a∣−(ln∣x∣−ln∣b∣) ln∣a∣=ln∣x∣−ln∣a∣−ln∣x∣+ln∣b∣ 2∙ln∣a∣=ln∣b∣ e^(ln∣a^2 ∣) =e^(ln∣b∣) a^2 =b if a^2 =b then a=((x−a)/(x−b)) a∙(x−b)=(x−a)⇔x−b≠0 a∙x−a∙b=x−a a∙x−x=a∙b−a x(a−1)=a∙a^2 −a x=((a^3 −a)/(a−1))⇔a−1≠0 x=((a(a^2 −1))/(a−1)) x=((a∙(a−1)∙(a+1))/(a−1)) x=a(a+1) x=b+a answer: x=a^2 +a or x=b+a, and x,a,b ∉ C so now the question is: what if x,a,b ∈ C?

$$ \\ $$$${a}=\frac{{x}−{a}}{{x}−{b}}\:{solve}\:{for}\:{x} \\ $$$$\mathrm{did}\:\mathrm{i}\:\mathrm{make}\:\mathrm{anything}\:\mathrm{wrong}\:\mathrm{in}\:\mathrm{the}\:\mathrm{following}? \\ $$$$ \\ $$$$\mathrm{starpoint}: \\ $$$${ln}\mid{a}\mid={ln}\mid\frac{{x}−{a}}{{x}−{b}}\mid \\ $$$${ln}\mid{a}\mid={ln}\mid{x}−{a}\mid−{ln}\mid{x}−{b}\mid \\ $$$${ln}\mid{a}\mid={ln}\mid\frac{{x}}{{a}}\mid−{ln}\mid\frac{{x}}{{b}}\mid \\ $$$${ln}\mid{a}\mid={ln}\mid{x}\mid−{ln}\mid{a}\mid−\left({ln}\mid{x}\mid−{ln}\mid{b}\mid\right) \\ $$$${ln}\mid{a}\mid={ln}\mid{x}\mid−{ln}\mid{a}\mid−{ln}\mid{x}\mid+{ln}\mid{b}\mid \\ $$$$\mathrm{2}\centerdot{ln}\mid{a}\mid={ln}\mid{b}\mid \\ $$$${e}^{{ln}\mid{a}^{\mathrm{2}} \mid} ={e}^{{ln}\mid{b}\mid} \\ $$$${a}^{\mathrm{2}} ={b} \\ $$$$\mathrm{if}\:{a}^{\mathrm{2}} ={b}\:\mathrm{then} \\ $$$${a}=\frac{{x}−{a}}{{x}−{b}} \\ $$$${a}\centerdot\left({x}−{b}\right)=\left({x}−{a}\right)\Leftrightarrow{x}−{b}\neq\mathrm{0} \\ $$$${a}\centerdot{x}−{a}\centerdot{b}={x}−{a} \\ $$$${a}\centerdot{x}−{x}={a}\centerdot{b}−{a} \\ $$$${x}\left({a}−\mathrm{1}\right)={a}\centerdot{a}^{\mathrm{2}} −{a} \\ $$$${x}=\frac{{a}^{\mathrm{3}} −{a}}{{a}−\mathrm{1}}\Leftrightarrow{a}−\mathrm{1}\neq\mathrm{0} \\ $$$${x}=\frac{{a}\left({a}^{\mathrm{2}} −\mathrm{1}\right)}{{a}−\mathrm{1}} \\ $$$${x}=\frac{{a}\centerdot\left({a}−\mathrm{1}\right)\centerdot\left({a}+\mathrm{1}\right)}{{a}−\mathrm{1}} \\ $$$${x}={a}\left({a}+\mathrm{1}\right) \\ $$$${x}={b}+{a} \\ $$$$ \\ $$$$\mathrm{answer}: \\ $$$${x}={a}^{\mathrm{2}} +{a}\:{or}\:{x}={b}+{a},\:\mathrm{and}\:{x},{a},{b}\:\notin\:\mathbb{C} \\ $$$$ \\ $$$$\mathrm{so}\:\mathrm{now}\:\mathrm{the}\:\mathrm{question}\:\mathrm{is}:\:\mathrm{what}\:\mathrm{if}\:{x},{a},{b}\:\in\:\mathbb{C}? \\ $$$$ \\ $$

Question Number 191250 Answers: 0 Comments: 4

Question Number 191243 Answers: 2 Comments: 1

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