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Question Number 141643 Answers: 1 Comments: 0

Question Number 141642 Answers: 0 Comments: 0

Question Number 141640 Answers: 1 Comments: 0

Question Number 141635 Answers: 0 Comments: 0

solve the differential equation (PDE), z((∂z/∂x)−(∂z/∂y))=z^2 +(x+y)^2 .

$${solve}\:{the}\:{differential}\:{equation}\:\left({PDE}\right), \\ $$$${z}\left(\frac{\partial{z}}{\partial{x}}−\frac{\partial{z}}{\partial{y}}\right)={z}^{\mathrm{2}} +\left({x}+{y}\right)^{\mathrm{2}} . \\ $$

Question Number 141633 Answers: 0 Comments: 0

1<a<b ,prove that : b^n = Σ_(k=0) ^n (−1)^k C_n ^k a^((ln(Σ_(p=0) ^(n−k) C_(n−k) ^p a^(n−p) b^p ))/(ln(a)))

$$\mathrm{1}<\mathrm{a}<\mathrm{b}\:,\mathrm{prove}\:\mathrm{that}\:: \\ $$$${b}^{{n}} \:=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{{n}} ^{{k}} \:{a}^{\frac{{ln}\left(\underset{{p}=\mathrm{0}} {\overset{{n}−{k}} {\sum}}\mathrm{C}_{{n}−{k}} ^{{p}} {a}^{{n}−{p}} {b}^{{p}} \right)}{{ln}\left({a}\right)}} \\ $$

Question Number 141632 Answers: 0 Comments: 0

Let f(x)=((sin(x))/x) , prove that : Σ_(n=0) ^∞ [ f(nπ+α)+f(nπ−α) ]= 1+f(α)

$$\mathrm{Let}\:{f}\left({x}\right)=\frac{{sin}\left({x}\right)}{{x}}\:,\:\mathrm{prove}\:\mathrm{that}\:: \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\:{f}\left({n}\pi+\alpha\right)+{f}\left({n}\pi−\alpha\right)\:\right]=\:\mathrm{1}+{f}\left(\alpha\right) \\ $$

Question Number 141669 Answers: 1 Comments: 0

Question Number 141628 Answers: 0 Comments: 3

Question Number 141627 Answers: 2 Comments: 0

Question Number 141623 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (∫_0 ^1 (x^n /(1+x))dx)^2 =ln 2

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}\right)^{\mathrm{2}} ={ln}\:\mathrm{2} \\ $$

Question Number 141614 Answers: 1 Comments: 2

Σ(1/(k+1))C_n ^k .

$$\Sigma\frac{\mathrm{1}}{{k}+\mathrm{1}}{C}_{{n}} ^{{k}} \:.\: \\ $$

Question Number 142218 Answers: 1 Comments: 0

Straight line lx+my=1 is tangent to the curve (ax)^n +(by)^n =1 Prove that ((l/a))^(n/(n−1)) +((m/b))^(n/(n−1)) =1

$${Straight}\:{line}\:{lx}+{my}=\mathrm{1}\:\:{is}\:{tangent}\:{to}\:{the}\:{curve}\:\left({ax}\right)^{{n}} +\left({by}\right)^{{n}} =\mathrm{1} \\ $$$${Prove}\:{that}\:\left(\frac{{l}}{{a}}\right)^{\frac{{n}}{{n}−\mathrm{1}}} +\left(\frac{{m}}{{b}}\right)^{\frac{{n}}{{n}−\mathrm{1}}} =\mathrm{1} \\ $$

Question Number 141599 Answers: 1 Comments: 1

∫_( 1) ^( 3) ∫_( − 1) ^( 1) ∫_( 0) ^( 2) (x + 2y − z) dx dy dz

$$\int_{\:\mathrm{1}} ^{\:\mathrm{3}} \:\int_{\:−\:\mathrm{1}} ^{\:\mathrm{1}} \int_{\:\mathrm{0}} ^{\:\mathrm{2}} \:\:\left(\mathrm{x}\:\:\:+\:\:\mathrm{2y}\:\:\:−\:\:\:\mathrm{z}\right)\:\mathrm{dx}\:\mathrm{dy}\:\mathrm{dz} \\ $$

Question Number 141598 Answers: 1 Comments: 0

Question Number 141612 Answers: 0 Comments: 0

Σ_(k=1) ^(n−1) (((−1)^(k+1) C_(k−1) ^( n−2) )/((k+1)^x ))=?

$$\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} {C}_{{k}−\mathrm{1}} ^{\:{n}−\mathrm{2}} }{\left({k}+\mathrm{1}\right)^{{x}} }=? \\ $$

Question Number 141611 Answers: 0 Comments: 1

Evaluate: ∫log(ex^2 )^x^(logx)

$$\:\: \\ $$$$\:\:{Evaluate}:\: \\ $$$$\:\:\:\int{log}\left({ex}^{\mathrm{2}} \right)^{{x}^{{logx}} } \\ $$

Question Number 141608 Answers: 1 Comments: 1

Question Number 141594 Answers: 1 Comments: 0

Question Number 141586 Answers: 1 Comments: 0

Find the value of x for which the following functions are undefined (1) g:x→((5x+1)/2) (2) f:x→3x^2 −5x+1

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{functions}\:\mathrm{are}\:\mathrm{undefined}\: \\ $$$$\left(\mathrm{1}\right)\:\mathrm{g}:\mathrm{x}\rightarrow\frac{\mathrm{5x}+\mathrm{1}}{\mathrm{2}} \\ $$$$\left(\mathrm{2}\right)\:\mathrm{f}:\mathrm{x}\rightarrow\mathrm{3x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{1}\: \\ $$

Question Number 141585 Answers: 1 Comments: 0

If n is a multiple of 4 and i=(√(−1)) , find the sum of the series S=1+2i+3i^2 +......+(n+1)i^n

$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{multiple}\:\mathrm{of}\:\mathrm{4}\:\mathrm{and}\:{i}=\sqrt{−\mathrm{1}}\:,\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{series} \\ $$$$\:\:\:{S}=\mathrm{1}+\mathrm{2}{i}+\mathrm{3}{i}^{\mathrm{2}} +......+\left({n}+\mathrm{1}\right){i}^{{n}} \: \\ $$

Question Number 141581 Answers: 1 Comments: 1

Σ_(n=0) ^∞ ∫_0 ^1 (x^n /(1+x))dx=?

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{1}+{x}}{dx}=? \\ $$

Question Number 141580 Answers: 0 Comments: 0

i am finding it difficult to form a pde from the function f(x+y+z, x^2 +y^2 +z^2 ).

$${i}\:{am}\:{finding}\:{it}\:{difficult}\:{to}\:{form}\:{a}\:{pde} \\ $$$${from}\:{the}\:{function}\:{f}\left({x}+{y}+{z},\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right). \\ $$

Question Number 141578 Answers: 4 Comments: 0

∫(√(1+(1/x)))dx=?

$$\int\sqrt{\mathrm{1}+\frac{\mathrm{1}}{{x}}}{dx}=? \\ $$

Question Number 141577 Answers: 0 Comments: 0

((1/(1∙2∙3)))^2 +((1/(2∙3∙4)))^2 +((1/(3∙4∙5)))^2 +...=(1/(16))(4π^2 −39)

$$\left(\frac{\mathrm{1}}{\mathrm{1}\centerdot\mathrm{2}\centerdot\mathrm{3}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}\right)^{\mathrm{2}} +\left(\frac{\mathrm{1}}{\mathrm{3}\centerdot\mathrm{4}\centerdot\mathrm{5}}\right)^{\mathrm{2}} +...=\frac{\mathrm{1}}{\mathrm{16}}\left(\mathrm{4}\pi^{\mathrm{2}} −\mathrm{39}\right) \\ $$

Question Number 141574 Answers: 0 Comments: 0

Question Number 141570 Answers: 1 Comments: 0

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