Question and Answers Forum

AllQuestion and Answers: Page 1125

Question Number 104939 Answers: 0 Comments: 0

Σ_(n=0) ^∞ (((−1)^(n+1) )/((n+1)!(2n+1))) = ((√π)/2)

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\left({n}+\mathrm{1}\right)!\left(\mathrm{2}{n}+\mathrm{1}\right)}\:=\:\frac{\sqrt{\pi}}{\mathrm{2}}\: \\ $$

Question Number 104910 Answers: 0 Comments: 0

∫_0 ^1 (1/(√(1+x^3 )))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\sqrt{\mathrm{1}+{x}^{\mathrm{3}} }}{dx} \\ $$

Question Number 104905 Answers: 3 Comments: 0

Question Number 104904 Answers: 3 Comments: 0

4cos^2 x sin x −2sin^2 x = 3sin x where −(π/2)≤x≤(π/2)

$$\mathrm{4cos}\:^{\mathrm{2}} {x}\:\mathrm{sin}\:{x}\:−\mathrm{2sin}\:^{\mathrm{2}} {x}\:=\:\mathrm{3sin}\:{x} \\ $$$${where}\:−\frac{\pi}{\mathrm{2}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}} \\ $$

Question Number 104899 Answers: 2 Comments: 0

lim_(x→0) ((cos (sin x)−cos (x))/x^4 ) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:\left(\mathrm{sin}\:{x}\right)−\mathrm{cos}\:\left({x}\right)}{{x}^{\mathrm{4}} }\:?\: \\ $$

Question Number 104895 Answers: 2 Comments: 0

1) decompose the fraction F(x) =(1/(x^3 (x+1)^4 )) 2) find the sumA = Σ_(n=1) ^∞ (1/(n^3 (n+1)^4 )) and B =Σ_(n=1) ^(∞ ) (((−1)^n )/(n^3 (n+1)^4 )) 3) what is the value of Σ_(n=0) ^∞ (1/((n+1)^4 (2n+1)^3 )) ?

$$\left.\mathrm{1}\right)\:\mathrm{decompose}\:\mathrm{the}\:\mathrm{fraction}\:\:\mathrm{F}\left(\mathrm{x}\right)\:=\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} \left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{sumA}\:=\:\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{3}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{4}} }\:\:\mathrm{and}\:\mathrm{B}\:=\sum_{\mathrm{n}=\mathrm{1}} ^{\infty\:} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{3}\right)\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\sum_{\mathrm{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{4}} \left(\mathrm{2n}+\mathrm{1}\right)^{\mathrm{3}} }\:? \\ $$

Question Number 104894 Answers: 1 Comments: 0

prove that cos^6 a −sin^6 a = cos 2a (1−(1/4)sin^2 2a)

$${prove}\:{that}\:\mathrm{cos}\:^{\mathrm{6}} {a}\:−\mathrm{sin}\:^{\mathrm{6}} {a}\:=\: \\ $$$$\mathrm{cos}\:\mathrm{2}{a}\:\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4}}\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{a}\right)\: \\ $$

Question Number 104893 Answers: 1 Comments: 0

(d^2 y/dx^2 ) + tan x (dy/dx) = sec x + cot x

$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{tan}\:{x}\:\frac{{dy}}{{dx}}\:=\:\mathrm{sec}\:{x}\:+\:\mathrm{cot}\:{x} \\ $$

Question Number 104891 Answers: 1 Comments: 0

1) decompose the fraction F(x) =(1/(x^3 (x+1)^3 )) 2) find the sum Σ_(n=1) ^∞ (((−1)^n )/(n^3 (n+1)^3 ))

$$\left.\mathrm{1}\right)\:\mathrm{decompose}\:\mathrm{the}\:\mathrm{fraction}\:\mathrm{F}\left(\mathrm{x}\right)\:=\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} \left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\sum_{\mathrm{n}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{3}} \left(\mathrm{n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Question Number 104890 Answers: 1 Comments: 0

lim_(n→∞) (1/n)Σ_(k=1) ^n [1+(k^3 /n^3 )]^(−(1/2))

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left[\mathrm{1}+\frac{\mathrm{k}^{\mathrm{3}} }{\mathrm{n}^{\mathrm{3}} }\right]^{−\frac{\mathrm{1}}{\mathrm{2}}} \\ $$

Question Number 104973 Answers: 1 Comments: 1

2(√3) + i is a cubic root for 18(√3) + 35i find the other 2 cubic roots

$$\mathrm{2}\sqrt{\mathrm{3}}\:+\:\boldsymbol{{i}}\:\:\:{is}\:{a}\:{cubic}\:{root}\:{for} \\ $$$$\mathrm{18}\sqrt{\mathrm{3}}\:+\:\mathrm{35}\boldsymbol{{i}}\: \\ $$$$\boldsymbol{{find}}\:\:\boldsymbol{{the}}\:\boldsymbol{{other}}\:\mathrm{2}\:\boldsymbol{{cubic}}\:\boldsymbol{{roots}} \\ $$

Question Number 104980 Answers: 1 Comments: 3

Question Number 104877 Answers: 1 Comments: 0

find x

$$ \\ $$$${find}\:{x} \\ $$

Question Number 104876 Answers: 0 Comments: 0

A ballot box contains 7 balls(3 are black). we draw successively and reputing(inside) 5 balls. What is the number of possibilities to have one black ball?

$${A}\:{ballot}\:{box}\:{contains}\:\mathrm{7}\:{balls}\left(\mathrm{3}\:{are}\:\right. \\ $$$$\left.{black}\right).\:{we}\:{draw}\:{successively}\:{and} \\ $$$${reputing}\left({inside}\right)\:\mathrm{5}\:{balls}. \\ $$$${What}\:{is}\:{the}\:{number}\:{of}\:{possibilities}\: \\ $$$${to}\:{have}\:{one}\:{black}\:{ball}? \\ $$

Question Number 104984 Answers: 0 Comments: 1