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Question Number 36784 Answers: 0 Comments: 0

Prove that Σ_(r=0) ^n r ((n),(r) )^2 = n (((2n − 1)),(( n − 1)) )

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\underset{{r}=\mathrm{0}} {\overset{{n}} {\sum}}\:{r}\:\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix}^{\mathrm{2}} \:=\:{n}\:\begin{pmatrix}{\mathrm{2}{n}\:−\:\mathrm{1}}\\{\:\:{n}\:−\:\mathrm{1}}\end{pmatrix} \\ $$

Question Number 36771 Answers: 0 Comments: 3

i have a suggestion...pls request members of the forum to post four to five question so that we get time to solve them...there is flood of questions...so little time to see all post pls give comment if you agree wkth it...

$${i}\:{have}\:{a}\:{suggestion}...{pls}\:{request}\:{members} \\ $$$${of}\:{the}\:{forum}\:{to}\:{post}\:{four}\:{to}\:{five}\:{question} \\ $$$${so}\:{that}\:{we}\:{get}\:{time}\:{to}\:{solve}\:{them}...{there}\:{is}\: \\ $$$${flood}\:{of}\:{questions}...{so}\:{little}\:{time}\:{to}\:{see}\:{all}\:{post} \\ $$$${pls}\:{give}\:{comment}\:{if}\:{you}\:{agree}\:{wkth}\:{it}... \\ $$

Question Number 36762 Answers: 1 Comments: 2

find A_n = ∫_0 ^(π/4) (cosx +sinx)^n dx.

$${find}\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\left({cosx}\:+{sinx}\right)^{{n}} \:{dx}. \\ $$

Question Number 36755 Answers: 1 Comments: 4

let f(a) = ∫_0 ^1 e^t ln(1+ e^(−at) )dt with a≥0 1) find f(a) 2) calculate f^′ (a) 3) find the value of ∫_0 ^1 e^t ln(1+e^(−3t) )dt .

$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{{t}} {ln}\left(\mathrm{1}+\:{e}^{−{at}} \right){dt}\:\:{with}\:{a}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{f}^{'} \left({a}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{t}} {ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{t}} \right){dt}\:. \\ $$

Question Number 36754 Answers: 1 Comments: 1

calculate ∫_1 ^(+∞) (dx/(x^2 (√(4+x^2 )))) .

$${calculate}\:\:\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\:\frac{{dx}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}+{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 36753 Answers: 1 Comments: 2

find I_n = ∫_0 ^1 x^n arctan(x)dx .

$${find}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{x}^{{n}} \:{arctan}\left({x}\right){dx}\:. \\ $$

Question Number 36752 Answers: 1 Comments: 4

find ∫ (dx/(arcsinx(√(1−x^2 )))) .

$${find}\:\:\int\:\:\:\frac{{dx}}{{arcsinx}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:. \\ $$

Question Number 36751 Answers: 0 Comments: 0

let f(x)= Σ_(n=1) ^∞ x^n^2 with x∈]−1,1[ prove that f(x) ∼ ((√π)/(2(√(−ln(x))))) (x →1^− )

$$\left.{let}\:\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{x}^{{n}^{\mathrm{2}} } \:\:\:{with}\:\:{x}\in\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$${prove}\:{that}\:\:{f}\left({x}\right)\:\sim\:\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{−{ln}\left({x}\right)}}\:\left({x}\:\rightarrow\mathrm{1}^{−} \right) \\ $$

Question Number 36750 Answers: 0 Comments: 0

let f(t)=Σ_(n≥1) (−1)^n ln{1+ (t^2 /(n(1+t^2 )))} 1) study the simple and uniform convergence of Σ f_n 2)study the continuity of f 3) prove that lim_(t→+∞) f(t)=ln((2/π)) .

$${let}\:{f}\left({t}\right)=\sum_{{n}\geqslant\mathrm{1}} \:\left(−\mathrm{1}\right)^{{n}} {ln}\left\{\mathrm{1}+\:\frac{{t}^{\mathrm{2}} }{{n}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}\right\} \\ $$$$\left.\mathrm{1}\right)\:{study}\:{the}\:{simple}\:\:{and}\:{uniform}\:{convergence} \\ $$$${of}\:\Sigma\:{f}_{{n}} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continuity}\:{of}\:{f} \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:{lim}_{{t}\rightarrow+\infty} \:{f}\left({t}\right)={ln}\left(\frac{\mathrm{2}}{\pi}\right)\:. \\ $$

Question Number 36748 Answers: 0 Comments: 0

let f(x)= Σ_(n=1) ^∞ (((−1)^(n−1) )/(ln(nx))) 1) give D_f and study f on]1,+∞[ 2)study the continjity of f and calculate lim _(x→1) f(x) and lim_(x→+∞) f(x). 3) prove that f is C^1 on ]1,+∞[ .

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{ln}\left({nx}\right)} \\ $$$$\left.\mathrm{1}\left.\right)\:{give}\:{D}_{{f}} \:\:{and}\:{study}\:{f}\:{on}\right]\mathrm{1},+\infty\left[\right. \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{continjity}\:{of}\:{f}\:{and}\:{calculate} \\ $$$${lim}\:_{{x}\rightarrow\mathrm{1}} {f}\left({x}\right)\:{and}\:{lim}_{{x}\rightarrow+\infty} {f}\left({x}\right). \\ $$$$\left.\mathrm{3}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]\mathrm{1},+\infty\left[\:.\right. \\ $$

Question Number 36747 Answers: 0 Comments: 1

let f(x)= Σ_(n=1) ^∞ ((sin(nx))/n) x^n 1) prove that f is C^1 on ]−1,1[ 2)calculate f^′ (x) and prove that f(x)=arctan( ((xsinx)/(1−x cosx)))

$${let}\:{f}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{sin}\left({nx}\right)}{{n}}\:{x}^{{n}} \\ $$$$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{1}} \:{on}\:\right]−\mathrm{1},\mathrm{1}\left[\right. \\ $$$$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({x}\right)\:{and}\:{prove}\:{that} \\ $$$${f}\left({x}\right)={arctan}\left(\:\frac{{xsinx}}{\mathrm{1}−{x}\:{cosx}}\right) \\ $$

Question Number 36746 Answers: 0 Comments: 0

solve the d.e. y^(′′) (t) +y(t)=Σ_(n=0) ^N a_n cos(nt)

$${solve}\:{the}\:{d}.{e}.\:{y}^{''} \left({t}\right)\:+{y}\left({t}\right)=\sum_{{n}=\mathrm{0}} ^{{N}} \:{a}_{{n}} {cos}\left({nt}\right) \\ $$

Question Number 36745 Answers: 0 Comments: 0

solve the d.e y^(′′) (t) +y(t)=cos(nt)

$${solve}\:{the}\:{d}.{e}\:\:{y}^{''} \left({t}\right)\:+{y}\left({t}\right)={cos}\left({nt}\right) \\ $$

Question Number 36744 Answers: 0 Comments: 0

let f(x)=Σ_(n=1) ^∞ (1/n) cos^n (x)sin(nx) 1)prove the convergence of this serie 2)prove that f is C^2 on R −{kπ,k∈Z}and calculate f^′ (x) 3) give a exprrssion of f.

$${let}\:{f}\left({x}\right)=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}}\:{cos}^{{n}} \left({x}\right){sin}\left({nx}\right) \\ $$$$\left.\mathrm{1}\right){prove}\:{the}\:{convergence}\:{of}\:{this}\:{serie} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:{C}^{\mathrm{2}} \:{on}\:{R}\:−\left\{{k}\pi,{k}\in{Z}\right\}{and} \\ $$$${calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{3}\right)\:{give}\:{a}\:{exprrssion}\:{of}\:{f}. \\ $$

Question Number 36742 Answers: 0 Comments: 1

study the convergence of Σ_(n=1) ^∞ (−1)^n ln(1+ (1/(n(1+x)))).

$${study}\:{the}\:{convergence}\:{of}\: \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} {ln}\left(\mathrm{1}+\:\frac{\mathrm{1}}{{n}\left(\mathrm{1}+{x}\right)}\right). \\ $$

Question Number 36741 Answers: 1 Comments: 1

calculate S(x)=Σ_(n=0) ^∞ ((sin(nx))/(n!))

$${calculate}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{n}!} \\ $$

Question Number 36739 Answers: 0 Comments: 3

If x = a(1 − cosθ)i + asinθ j find the resultant of x in its simplest form.

$$\mathrm{If}\:\:\:\mathrm{x}\:=\:\mathrm{a}\left(\mathrm{1}\:−\:\mathrm{cos}\theta\right)\mathrm{i}\:\:+\:\:\mathrm{asin}\theta\:\mathrm{j} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{resultant}\:\mathrm{of}\:\mathrm{x}\:\mathrm{in}\:\mathrm{its}\:\mathrm{simplest}\:\mathrm{form}. \\ $$

Question Number 36738 Answers: 2 Comments: 3

(1) ∫(dα/((1+sin 2α)^2 ))= (2) ∫(dβ/((1+cos 2β)^2 ))= (3) ∫(dγ/((1+sin 2γ)(1+cos 2γ)))=