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Question Number 33411 Answers: 0 Comments: 1

Given that u and v are real valued functions in x ,then (d/dx)((u/v)) is equal to?

$$\:\mathrm{Given}\:\mathrm{that}\:{u}\:\mathrm{and}\:{v}\:\mathrm{are}\:\mathrm{real}\:\mathrm{valued} \\ $$$$\mathrm{functions}\:\mathrm{in}\:{x}\:,\mathrm{then}\:\frac{{d}}{{dx}}\left(\frac{{u}}{{v}}\right)\:{is}\:{equal}\:{to}? \\ $$

Question Number 33410 Answers: 1 Comments: 1

please is there any general way for calculating the error or uncertainty in g when m=((4π^2 )/g) where m=slope and g=acceleration due to gravity please help

$${please}\:{is}\:{there}\:{any}\:{general}\:{way}\:{for} \\ $$$${calculating}\:{the}\:{error}\:{or}\:{uncertainty} \\ $$$${in}\:{g}\:{when} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{m}=\frac{\mathrm{4}\pi^{\mathrm{2}} }{{g}}\:{where}\:{m}={slope}\:{and} \\ $$$${g}={acceleration}\:{due}\:{to}\:{gravity} \\ $$$$ \\ $$$$ \\ $$$${please}\:{help} \\ $$

Question Number 33407 Answers: 1 Comments: 0

if y=x! find dy/dx

$${if}\:{y}={x}!\:{find}\:{dy}/{dx} \\ $$

Question Number 33406 Answers: 0 Comments: 4

Find the half derivative of y=ln x

$${Find}\:{the}\:{half}\:{derivative}\:{of}\:{y}=\mathrm{ln}\:{x} \\ $$

Question Number 33400 Answers: 0 Comments: 4

Find out electric field on an axial position due to a ring having linear charge density 𝛌= λ_0 cos θ .

$$\boldsymbol{{Find}}\:{out}\:{electric}\:{field}\:{on}\:{an}\:{axial}\: \\ $$$${position}\:{due}\:{to}\:{a}\:{ring}\:{having}\:{linear} \\ $$$${charge}\:{density}\:\boldsymbol{\lambda}=\:\lambda_{\mathrm{0}} \:\mathrm{cos}\:\theta\:. \\ $$

Question Number 33375 Answers: 0 Comments: 2

If f:R → R is an odd function such that : a) f(1+x) = 1+f(x) . b) x^2 f((1/x)) = f(x) , x≠0. Then find f(x) ?

$${If}\:{f}:{R}\:\rightarrow\:{R}\:{is}\:{an}\:\boldsymbol{{odd}}\:{function}\:{such} \\ $$$${that}\:: \\ $$$$\left.{a}\right)\:{f}\left(\mathrm{1}+{x}\right)\:=\:\mathrm{1}+{f}\left({x}\right)\:. \\ $$$$\left.{b}\right)\:{x}^{\mathrm{2}} \:{f}\left(\frac{\mathrm{1}}{{x}}\right)\:=\:{f}\left({x}\right)\:,\:{x}\neq\mathrm{0}. \\ $$$${Then}\:{find}\:\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\:? \\ $$

Question Number 33369 Answers: 1 Comments: 0

Prove that gcd( gcd(A,B),gcd(B,C),gcd(C,A) ) =gcd(A,B,C)

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\mathrm{gcd}\left(\:\:\mathrm{gcd}\left(\mathrm{A},\mathrm{B}\right),\mathrm{gcd}\left(\mathrm{B},\mathrm{C}\right),\mathrm{gcd}\left(\mathrm{C},\mathrm{A}\right)\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{gcd}\left(\mathrm{A},\mathrm{B},\mathrm{C}\right) \\ $$

Question Number 33363 Answers: 0 Comments: 3

Question Number 33362 Answers: 0 Comments: 1

calculate by residus theorem I = ∫_(−∞) ^(+∞) ((cos(πx))/((1+x +x^2 )))dx .

$${calculate}\:{by}\:{residus}\:{theorem} \\ $$$${I}\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\pi{x}\right)}{\left(\mathrm{1}+{x}\:+{x}^{\mathrm{2}} \right)}{dx}\:. \\ $$

Question Number 33359 Answers: 0 Comments: 0

let consider the serie Σ_(n≥1) sin((1/(√n)))x^n 1) find the radius of convergence 2)study the convergence at −R and R 3) let S(x)its sum study the continuity of S 4) prove that (1−x)_(x→1^− ) S(x)→0

$${let}\:{consider}\:{the}\:{serie}\:\sum_{{n}\geqslant\mathrm{1}} {sin}\left(\frac{\mathrm{1}}{\sqrt{{n}}}\right){x}^{{n}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{radius}\:{of}\:{convergence} \\ $$$$\left.\mathrm{2}\right){study}\:{the}\:{convergence}\:{at}\:−{R}\:{and}\:{R} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{S}\left({x}\right){its}\:{sum}\:{study}\:{the}\:{continuity} \\ $$$${of}\:{S} \\ $$$$\left.\mathrm{4}\right)\:{prove}\:{that}\:\left(\mathrm{1}−{x}\right)_{{x}\rightarrow\mathrm{1}^{−} } {S}\left({x}\right)\rightarrow\mathrm{0} \\ $$

Question Number 33358 Answers: 0 Comments: 0

prove that Σ_(p=1) ^∞ (z^p /(1+z^p )) =Σ_(q=1) ^∞ (−1)^(q−1) (z^q /(1−z^q ))

$${prove}\:{that}\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\frac{{z}^{{p}} }{\mathrm{1}+{z}^{{p}} }\:=\sum_{{q}=\mathrm{1}} ^{\infty} \:\left(−\mathrm{1}\right)^{{q}−\mathrm{1}} \frac{{z}^{{q}} }{\mathrm{1}−{z}^{{q}} } \\ $$

Question Number 33357 Answers: 0 Comments: 0

find the rsdius of convergence for the serie Σ_(n=1) ^∞ (1 +(1/n))^n^2 x^n

$${find}\:{the}\:{rsdius}\:{of}\:{convergence}\:{for} \\ $$$${the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}\:+\frac{\mathrm{1}}{{n}}\right)^{{n}^{\mathrm{2}} } \:{x}^{{n}} \: \\ $$$$ \\ $$

Question Number 33356 Answers: 0 Comments: 0

find the radius of Σ_(n≥0) ((n^2 +n)/(2^n +n!)) x^n

$${find}\:{the}\:{radius}\:{of}\:\sum_{{n}\geqslant\mathrm{0}} \frac{{n}^{\mathrm{2}} \:+{n}}{\mathrm{2}^{{n}} \:+{n}!}\:{x}^{{n}} \\ $$

Question Number 33355 Answers: 0 Comments: 0

let a≥1 find the radius of Σ_(n≥1) arc cos(1−(1/n^a ))z^n

$${let}\:{a}\geqslant\mathrm{1}\:{find}\:{the}\:{radius}\:{of} \\ $$$$\sum_{{n}\geqslant\mathrm{1}} {arc}\:{cos}\left(\mathrm{1}−\frac{\mathrm{1}}{{n}^{{a}} }\right){z}^{{n}} \\ $$

Question Number 33354 Answers: 0 Comments: 1

find the radius of Σ_(n≥1) ((ln(n))/(√(n^3 +n+1))) z^n

$${find}\:{the}\:{radius}\:{of}\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{{ln}\left({n}\right)}{\sqrt{{n}^{\mathrm{3}} \:+{n}+\mathrm{1}}}\:{z}^{{n}} \\ $$

Question Number 33353 Answers: 0 Comments: 1

let x∈]1,+∞[ andλ ∈[−1,1] give the integral ∫_0 ^∞ ((t^(x−1) e^(−t) )/(1−λe^(−t) )) dt at form of serie.

$$\left.{let}\:{x}\in\right]\mathrm{1},+\infty\left[\:{and}\lambda\:\in\left[−\mathrm{1},\mathrm{1}\right]\right. \\ $$$${give}\:{the}\:{integral}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} }{\mathrm{1}−\lambda{e}^{−{t}} }\:{dt} \\ $$$${at}\:{form}\:{of}\:{serie}. \\ $$

Question Number 33352 Answers: 0 Comments: 1

let give S(x)=Σ_(n≥0) (((−1)^n )/(√(x+n))) ,x>0 1)study the contnuity ,derivsbility,limits at 0^+ and +∞ 2) we give ∫_0 ^∞ e^(−t^2 ) dt =((√π)/2) .prove that ∀ x>0 S(x)=(1/(√π)) ∫_0 ^∞ (e^(−tx) /((√t)(1+e^(−t) )))dt .

$${let}\:{give}\:{S}\left({x}\right)=\sum_{{n}\geqslant\mathrm{0}} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\sqrt{{x}+{n}}}\:,{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{contnuity}\:,{derivsbility},{limits} \\ $$$${at}\:\mathrm{0}^{+} \:{and}\:+\infty \\ $$$$\left.\mathrm{2}\right)\:{we}\:{give}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{t}^{\mathrm{2}} } {dt}\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:.{prove}\:{that} \\ $$$$\forall\:{x}>\mathrm{0}\:\:{S}\left({x}\right)=\frac{\mathrm{1}}{\sqrt{\pi}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}} }{\sqrt{{t}}\left(\mathrm{1}+{e}^{−{t}} \right)}{dt}\:. \\ $$

Question Number 33351 Answers: 0 Comments: 1

prove that ∫_0 ^1 (((−lnx)^p )/(1+x^2 )) =p! Σ_(n=0) ^∞ (((−1)^n )/((2n+1)^(p+1) )) p integr.

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\left(−{lnx}\right)^{{p}} }{\mathrm{1}+{x}^{\mathrm{2}} }\:={p}!\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{{p}+\mathrm{1}} } \\ $$$${p}\:{integr}. \\ $$

Question Number 33350 Answers: 0 Comments: 1

prove that ∫_0 ^∞ ((cos(αx))/(chx))dx= 2 Σ_(n=0) ^∞ (−1)^n ((2n+1)/((2n+1)^2 +α^2 )) .

$${prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left(\alpha{x}\right)}{{chx}}{dx}= \\ $$$$\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \left(−\mathrm{1}\right)^{{n}} \:\frac{\mathrm{2}{n}+\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} \:+\alpha^{\mathrm{2}} }\:. \\ $$

Question Number 33349 Answers: 0 Comments: 1

prove that ∫_0 ^∞ x(x−ln(e^x −1))dx=Σ_(n=1) ^∞ (1/n^3 )

$${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:{x}\left({x}−{ln}\left({e}^{{x}} −\mathrm{1}\right)\right){dx}=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} } \\ $$

Question Number 33348 Answers: 0 Comments: 0

let S_n = Σ_(k=1) ^∞ (((−1)^(k−1) )/k) and T_n = Σ_(k=1) ^∞ (((−1)^(k−1) )/(2k−1)) 1) calculate lim S_n and lim T_n (n→∞) 2)prove that Σ(S_n −ln2) and Σ(T_n −(π/4))converges and find its sum

$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{and} \\ $$$$\underset{{n}} {{T}}\:=\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{\mathrm{2}{k}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}\:{S}_{{n}} \:\:{and}\:{lim}\:{T}_{{n}} \left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\Sigma\left({S}_{{n}} −{ln}\mathrm{2}\right)\:{and} \\ $$$$\Sigma\left({T}_{{n}} \:−\frac{\pi}{\mathrm{4}}\right){converges}\:{and}\:{find}\:{its}\:{sum} \\ $$$$ \\ $$

Question Number 33347 Answers: 0 Comments: 0

let f_n (x)= nx^2 e^(−x(√n)) , x≥0 1)study the simple convergence of Σ f_n (x{ 2) study the uniform convergence of Σ f_n (x).

$${let}\:{f}_{{n}} \left({x}\right)=\:{nx}^{\mathrm{2}} \:{e}^{−{x}\sqrt{{n}}} \:\:\:,\:{x}\geqslant\mathrm{0} \\ $$$$\left.\mathrm{1}\right){study}\:{the}\:{simple}\:{convergence}\:{of} \\ $$$$\Sigma\:{f}_{{n}} \left({x}\left\{\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{uniform}\:{convergence}\:{of} \\ $$$$\Sigma\:{f}_{{n}} \left({x}\right). \\ $$

Question Number 33346 Answers: 0 Comments: 0

let a and b from R /a<b f [a,b]→C continje prove that ∀n ∈N ∫_a ^b (Π_(k=0) ^(n−1) (x+k))f(x)dx=0 ⇒ f=0

$${let}\:{a}\:{and}\:{b}\:{from}\:{R}\:/{a}<{b}\:\:{f}\:\:\left[{a},{b}\right]\rightarrow{C}\:{continje} \\ $$$${prove}\:{that}\:\:\forall{n}\:\in{N}\:\:\int_{{a}} ^{{b}} \left(\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({x}+{k}\right)\right){f}\left({x}\right){dx}=\mathrm{0} \\ $$$$\Rightarrow\:{f}=\mathrm{0} \\ $$

Question Number 33345 Answers: 0 Comments: 0

for x∈]0,+∞[ let ψ(x) = ((Γ^′ (x))/(Γ(x))) 1)prove that ψ(x) =−(1/x) −γ +x Σ_(n=1) ^∞ (1/(n(x+n))) 2)ptove that γ =−Γ^′ (1) 3) prove that ∫_0 ^∞ e^(−x) ln(x)dx =−γ .

$$\left.{for}\:{x}\in\right]\mathrm{0},+\infty\left[\:{let}\:\psi\left({x}\right)\:=\:\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)}\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\psi\left({x}\right)\:=−\frac{\mathrm{1}}{{x}}\:−\gamma\:+{x}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}\left({x}+{n}\right)} \\ $$$$\left.\mathrm{2}\right){ptove}\:{that}\:\gamma\:=−\Gamma^{'} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {ln}\left({x}\right){dx}\:=−\gamma\:. \\ $$

Question Number 33344 Answers: 0 Comments: 1

prove that ∀ α ∈]0,+∞[ lim_(n→∞) ∫_0 ^n (1−(x/n))^n x^(α−1) dx =Γ(α) .

$$\left.{prove}\:{that}\:\:\forall\:\alpha\:\in\right]\mathrm{0},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \:{x}^{\alpha−\mathrm{1}} {dx}\:=\Gamma\left(\alpha\right)\:. \\ $$

Question Number 33343 Answers: 0 Comments: 2

prove that ∀ α ∈]1,+∞[ lim_(n→∞) ∫_0 ^n (1+(x/n))^n e^(−αx) dx = (1/(α−1)) .

$$\left.{prove}\:{that}\:\forall\:\alpha\:\in\right]\mathrm{1},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} \:{e}^{−\alpha{x}} {dx}\:=\:\frac{\mathrm{1}}{\alpha−\mathrm{1}}\:. \\ $$

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