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Question Number 197639 Answers: 1 Comments: 1

show that (1/((1−z)^n )) = Σ_(k=1) ^∞ (((n+k−1)!)/(k!(n−1)!)) z^k

$${show}\:{that}\:\frac{\mathrm{1}}{\left(\mathrm{1}−{z}\right)^{{n}} }\:=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left({n}+{k}−\mathrm{1}\right)!}{{k}!\left({n}−\mathrm{1}\right)!}\:{z}^{{k}} \\ $$

Question Number 197637 Answers: 1 Comments: 3

Question Number 197636 Answers: 1 Comments: 0

Question Number 197635 Answers: 0 Comments: 1

Question Number 197634 Answers: 0 Comments: 0

Question Number 197629 Answers: 1 Comments: 0

∫_0 ^3 e^(tE(t)) dt

$$\int_{\mathrm{0}} ^{\mathrm{3}} {e}^{{tE}\left({t}\right)} \:{dt} \\ $$

Question Number 197623 Answers: 1 Comments: 1

x,y∈N 162 ∙ x^2 = y^3 min(x+y)=?

$$\mathrm{x},\mathrm{y}\in\mathbb{N} \\ $$$$\mathrm{162}\:\centerdot\:\mathrm{x}^{\mathrm{2}} \:=\:\mathrm{y}^{\mathrm{3}} \\ $$$$\mathrm{min}\left(\mathrm{x}+\mathrm{y}\right)=? \\ $$

Question Number 197622 Answers: 2 Comments: 0

Question Number 197619 Answers: 2 Comments: 1

Question Number 197618 Answers: 1 Comments: 0

Question Number 197670 Answers: 1 Comments: 4

Question Number 197610 Answers: 0 Comments: 0

Question Number 197609 Answers: 2 Comments: 0

((x+3)/(2022)) + ((x+2)/(2023)) + ((x+1)/(2024)) + (x/(2025)) = −4

$$\:\:\:\frac{{x}+\mathrm{3}}{\mathrm{2022}}\:+\:\frac{{x}+\mathrm{2}}{\mathrm{2023}}\:+\:\frac{{x}+\mathrm{1}}{\mathrm{2024}}\:+\:\frac{{x}}{\mathrm{2025}}\:=\:−\mathrm{4} \\ $$

Question Number 197595 Answers: 2 Comments: 1

Question Number 197575 Answers: 0 Comments: 0

find ∫_0 ^1 (tanx)^(1/n) dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({tanx}\right)^{\frac{\mathrm{1}}{{n}}} {dx} \\ $$

Question Number 197570 Answers: 0 Comments: 0

Question Number 197569 Answers: 0 Comments: 2

Question Number 197567 Answers: 1 Comments: 0

Question Number 197564 Answers: 1 Comments: 4

sir...number of 3 digit numbers which are divisible by a)3 b)4 c)6 d)7 e)8 f)9 g)11 when repetetion is 1)Allowwd 2)Not allowed.. kindly help me sir

$${sir}...{number}\:{of}\:\mathrm{3}\:{digit} \\ $$$${numbers}\:{which}\:{are}\:{divisible} \\ $$$${by}\: \\ $$$$\left.{a}\left.\right)\left.\mathrm{3}\left.\:\left.\:\left.{b}\left.\right)\mathrm{4}\:\:{c}\right)\mathrm{6}\:\:{d}\right)\mathrm{7}\:\:{e}\right)\mathrm{8}\:\:{f}\right)\mathrm{9}\:\:{g}\right)\mathrm{11} \\ $$$${when}\:{repetetion}\:{is} \\ $$$$\left.\mathrm{1}\left.\right){Allowwd}\:\:\mathrm{2}\right){Not}\:{allowed}.. \\ $$$${kindly}\:{help}\:{me}\:{sir} \\ $$

Question Number 197562 Answers: 0 Comments: 0

let f_n (x) = nsin^(2n+1) x cos x then the value of lim_(n→∞) ∫_0 ^(π/2) f_n (x) dx − ∫_0 ^(π/2) ( lim_(n→∞) f_n (x))dx = ?

$$\:\:\mathrm{let}\:\:\mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\:=\:\mathrm{nsin}^{\mathrm{2n}+\mathrm{1}} \mathrm{x}\:\mathrm{cos}\:\mathrm{x}\:\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\:\mathrm{dx}\:−\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \left(\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{f}_{\mathrm{n}} \left(\mathrm{x}\right)\right)\mathrm{dx}\:\:\:=\:\:?\: \\ $$

Question Number 197550 Answers: 1 Comments: 0

Calcul I=∫^( (π/2)) _( 0) ((ln(cost))/(1+sin^2 t))dt

$$\mathrm{Calcul}\:\:\:\mathrm{I}=\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{cost}\right)}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \mathrm{t}}\mathrm{dt} \\ $$

Question Number 197549 Answers: 0 Comments: 3

solve limits for functions f(x)=cos(sgn(1/x)) f(x)=sgn(cos(1/x)) Can someone help Thanks

$$ \\ $$$${solve}\:{limits}\:{for}\:{functions} \\ $$$${f}\left({x}\right)={cos}\left({sgn}\left(\mathrm{1}/{x}\right)\right) \\ $$$${f}\left({x}\right)={sgn}\left({cos}\left(\mathrm{1}/{x}\right)\right) \\ $$$$ \\ $$$${Can}\:{someone}\:{help} \\ $$$${Thanks} \\ $$

Question Number 197548 Answers: 1 Comments: 0

lim_(x→−∞) (((4x−(√(4x^2 +5)))/(2x−1)))^(bx) =?

$$\:\:\:\:\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{4x}−\sqrt{\mathrm{4x}^{\mathrm{2}} +\mathrm{5}}}{\mathrm{2x}−\mathrm{1}}\right)^{\mathrm{bx}} =?\: \\ $$

Question Number 197541 Answers: 2 Comments: 3

Montrer que x=((an+bm)/(m+n))

$$\boldsymbol{\mathrm{Montrer}}\:\boldsymbol{\mathrm{que}} \\ $$$$\:\:\:\:\:\boldsymbol{\mathrm{x}}=\frac{\boldsymbol{\mathrm{an}}+\boldsymbol{\mathrm{bm}}}{\boldsymbol{\mathrm{m}}+\boldsymbol{\mathrm{n}}} \\ $$

Question Number 197530 Answers: 2 Comments: 0

f(x)−(x^2 /2)f′′(x)=0 f(x)=?

$$\:\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\mathrm{f}''\left(\mathrm{x}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 197525 Answers: 2 Comments: 0

∫ ((tanx))^(1/n) dx

$$\int\:\sqrt[{{n}}]{{tanx}}\:{dx} \\ $$

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