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Question Number 31748    Answers: 0   Comments: 3

1)find the value of Σ_(n=0) ^∞ (x^(3n) /((3n)!)) 2) find Σ_(n=0) ^∞ (8^n /((3n)!)) .

$$\left.\mathrm{1}\right){find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{3}{n}} }{\left(\mathrm{3}{n}\right)!} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{8}^{{n}} }{\left(\mathrm{3}{n}\right)!}\:\:. \\ $$

Question Number 31747    Answers: 0   Comments: 1

let give ∣λ∣<1 and u_n = ∫_0 ^π ((cos(nx))/(1−2λ cosx +λ^2 )) prove that Σ_(n=0) ^∞ u_n is convergent and find its sum .

$${let}\:{give}\:\mid\lambda\mid<\mathrm{1}\:{and}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{cos}\left({nx}\right)}{\mathrm{1}−\mathrm{2}\lambda\:{cosx}\:+\lambda^{\mathrm{2}} } \\ $$$${prove}\:{that}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{sum}\:. \\ $$

Question Number 31743    Answers: 1   Comments: 0

Find the value of : Σ_(i=0) ^∞ Σ_(j=0) ^∞ Σ_(k=0) ^∞ (1/(3^i 3^j 3^k )). case 1: i≠j≠k. case 2: i<j<k.

$$\:{Find}\:{the}\:{value}\:{of}\::\:\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{j}=\mathrm{0}} {\overset{\infty} {\sum}}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{3}^{{i}} \mathrm{3}^{{j}} \mathrm{3}^{{k}} }. \\ $$$${case}\:\mathrm{1}:\:{i}\neq{j}\neq{k}. \\ $$$${case}\:\mathrm{2}:\:{i}<{j}<{k}. \\ $$

Question Number 31740    Answers: 1   Comments: 0

Consider a particle in a uniform charge electric field.If the particle has a charge of 2C an is placed 3m away from the charge flate. Calculate the work needed to move the 2C particle to a distance of 1m from the plate. Plzz help

$$\boldsymbol{\mathcal{C}{onsider}}\:\boldsymbol{{a}}\:\boldsymbol{{particle}}\:\boldsymbol{{in}}\:\boldsymbol{{a}}\:\boldsymbol{{uniform}} \\ $$$$\boldsymbol{{charge}}\:\boldsymbol{{electric}}\:\boldsymbol{{field}}.\boldsymbol{{If}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{particle}}\:\boldsymbol{{has}}\:\boldsymbol{{a}}\:\boldsymbol{{charge}}\:\boldsymbol{{of}}\:\:\mathrm{2}\boldsymbol{{C}} \\ $$$$\boldsymbol{{an}}\:\boldsymbol{{is}}\:\boldsymbol{{placed}}\:\mathrm{3}\boldsymbol{{m}}\:\boldsymbol{{away}}\:\boldsymbol{{from}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{charge}}\:\boldsymbol{{flate}}. \\ $$$$\boldsymbol{{Calculate}}\:\boldsymbol{{the}}\:\boldsymbol{{work}}\:\boldsymbol{{needed}} \\ $$$$\boldsymbol{{to}}\:\boldsymbol{{move}}\:\boldsymbol{{the}}\:\mathrm{2}\boldsymbol{{C}}\:\boldsymbol{{particle}} \\ $$$$\boldsymbol{{to}}\:\boldsymbol{{a}}\:\boldsymbol{{distance}}\:\boldsymbol{{of}}\:\mathrm{1}\boldsymbol{{m}}\:\boldsymbol{{from}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{plate}}. \\ $$$$ \\ $$$$\boldsymbol{{Plzz}}\:\boldsymbol{{help}} \\ $$$$ \\ $$

Question Number 31734    Answers: 1   Comments: 0

Question Number 31745    Answers: 1   Comments: 0

pls, why is cos^4 −sin^(4 ) = cos^2 −sin^2 ?

$$\mathrm{pls},\:\mathrm{why}\:\mathrm{is}\:\mathrm{cos}^{\mathrm{4}} −\mathrm{sin}^{\mathrm{4}\:} \:=\:\mathrm{cos}^{\mathrm{2}} −\mathrm{sin}^{\mathrm{2}} ? \\ $$

Question Number 31726    Answers: 1   Comments: 3

Let a_1 = (1/2) , a_(k+1) =a_k ^2 +a_k ∀ k≥ 1. then a_(101) is greater than a) 1 b) 2 c) 3 d) 4 .

$${Let}\:{a}_{\mathrm{1}} =\:\frac{\mathrm{1}}{\mathrm{2}}\:,\:{a}_{{k}+\mathrm{1}} ={a}_{{k}} ^{\mathrm{2}} +{a}_{{k}} \forall\:{k}\geqslant\:\mathrm{1}. \\ $$$${then}\:{a}_{\mathrm{101}} \:\:{is}\:{greater}\:{than} \\ $$$$\left.{a}\right)\:\mathrm{1}\: \\ $$$$\left.{b}\right)\:\mathrm{2} \\ $$$$\left.{c}\right)\:\mathrm{3} \\ $$$$\left.{d}\right)\:\mathrm{4}\:. \\ $$

Question Number 31719    Answers: 0   Comments: 3

Question Number 31717    Answers: 1   Comments: 0

Question Number 31715    Answers: 0   Comments: 0

Question Number 31714    Answers: 0   Comments: 0

Prove that Σ_1 ^(n−2) k ((( n−2)),(( k)) ) (((n+2)),((k+2)) ) = (n−2) (((2n−1)),((n−1)) )

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\underset{\mathrm{1}} {\overset{{n}−\mathrm{2}} {\sum}}{k}\begin{pmatrix}{\:{n}−\mathrm{2}}\\{\:\:{k}}\end{pmatrix}\begin{pmatrix}{{n}+\mathrm{2}}\\{{k}+\mathrm{2}}\end{pmatrix}\:=\:\left({n}−\mathrm{2}\right)\begin{pmatrix}{\mathrm{2}{n}−\mathrm{1}}\\{{n}−\mathrm{1}}\end{pmatrix} \\ $$

Question Number 31713    Answers: 0   Comments: 3

Given n ∈ N prove that Σ_(k=1) ^n k(n+1−k)= ((( n+2)),(( 3)) )

$$\mathrm{Given}\:{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}\left({n}+\mathrm{1}−{k}\right)=\begin{pmatrix}{\:{n}+\mathrm{2}}\\{\:\:\:\:\:\mathrm{3}}\end{pmatrix} \\ $$

Question Number 31712    Answers: 0   Comments: 0

let f convex function on [0, 2π] with f ′′(x) ≤ M . find values a and b so a≤∫_0 ^(2π) f(x)cos x dx ≤bM

$$\mathrm{let}\:{f}\:\mathrm{convex}\:\mathrm{function}\:\mathrm{on}\:\left[\mathrm{0},\:\mathrm{2}\pi\right] \\ $$$$\mathrm{with}\:{f}\:''\left({x}\right)\:\leqslant\:{M}\:. \\ $$$$\mathrm{find}\:\mathrm{values}\:{a}\:\mathrm{and}\:{b}\:\:\mathrm{so} \\ $$$${a}\leqslant\int_{\mathrm{0}} ^{\mathrm{2}\pi} {f}\left({x}\right)\mathrm{cos}\:{x}\:{dx}\:\leqslant{bM} \\ $$

Question Number 31711    Answers: 0   Comments: 0

Given p is primes and A={−(m/n)−p(n/m) ∣ m , n ∈N} find sup A

$$\mathrm{Given}\:{p}\:\mathrm{is}\:\mathrm{primes}\:\mathrm{and}\:\mathrm{A}=\left\{−\frac{{m}}{{n}}−{p}\frac{{n}}{{m}}\:\mid\:{m}\:,\:{n}\:\in\mathbb{N}\right\} \\ $$$$\mathrm{find}\:\mathrm{sup}\:\mathrm{A} \\ $$

Question Number 31709    Answers: 0   Comments: 0

Given sequence (y_n ) with y_1 =1 , y_(n+1) =(1/4)(y_n ^3 +y_n ^2 )−1 find lim_(x→∞) y_n

$$\mathrm{Given}\:\mathrm{sequence}\:\left({y}_{{n}} \right)\:\mathrm{with}\:{y}_{\mathrm{1}} =\mathrm{1}\:, \\ $$$${y}_{{n}+\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{4}}\left({y}_{{n}} ^{\mathrm{3}} +{y}_{{n}} ^{\mathrm{2}} \right)−\mathrm{1}\: \\ $$$$\mathrm{find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{y}_{{n}} \\ $$

Question Number 31708    Answers: 0   Comments: 0

Given sequence real function (f_n ) , f_n : [0, 2] → R ,with f_n (x)=(x^n /(1+x^n )) . n=1, 2, 3, ... a.Prove (f_n ) not uniformly convergent on [0, 2] b. Find lim_(x→∞) f_n (x) , x ∈ [0, 2]

$$\mathrm{Given}\:\mathrm{sequence}\:\mathrm{real}\:\mathrm{function}\:\left({f}_{{n}} \right)\:, \\ $$$${f}_{{n}} :\:\left[\mathrm{0},\:\mathrm{2}\right]\:\rightarrow\:\mathbb{R}\:,\mathrm{with}\: \\ $$$${f}_{{n}} \left({x}\right)=\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}} }\:\:.\:{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:... \\ $$$$\mathrm{a}.\mathrm{Prove}\:\left({f}_{{n}} \right)\:\mathrm{not}\:\mathrm{uniformly}\:\mathrm{convergent}\:\mathrm{on}\:\left[\mathrm{0},\:\mathrm{2}\right] \\ $$$$\mathrm{b}.\:\mathrm{Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}_{{n}} \left({x}\right)\:,\:{x}\:\in\:\left[\mathrm{0},\:\mathrm{2}\right] \\ $$

Question Number 31707    Answers: 0   Comments: 1

Find lim_(x→∞) Σ_(k=1) ^(n) (1/n)sin((2π)/n)

$$\mathrm{Find}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\Sigma}}\frac{\mathrm{1}}{{n}}\mathrm{sin}\frac{\mathrm{2}\pi}{{n}}\: \\ $$

Question Number 31706    Answers: 0   Comments: 1

Given θ_n = arc tan n , lim_(x→∞) θ_(n+1) −θ_n =

$$\mathrm{Given}\:\theta_{{n}} =\:{arc}\:\mathrm{tan}\:{n}\:,\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\theta_{{n}+\mathrm{1}} −\theta_{{n}} = \\ $$

Question Number 31705    Answers: 0   Comments: 0

Given sequence(x_n ) with 0<a=x_1 <x_2 =b x_(n+1) =x_(n+1) +x_n , n=1, 2, 3,... review sequence (r_n ) with r_n =(x_(n+1) /x_n ) , n=1, 2, 3,... a. Prove that 1<r_n <2 for n=2,3,4, ... b. Diverge or converge is the squence?

$$\mathrm{Given}\:\mathrm{sequence}\left({x}_{{n}} \right)\:\mathrm{with}\:\mathrm{0}<{a}={x}_{\mathrm{1}} <{x}_{\mathrm{2}} ={b} \\ $$$${x}_{{n}+\mathrm{1}} ={x}_{{n}+\mathrm{1}} +{x}_{{n}} \:,\:{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},... \\ $$$$\mathrm{review}\:\mathrm{sequence}\:\left({r}_{{n}} \right)\:\mathrm{with}\:{r}_{{n}} =\frac{{x}_{{n}+\mathrm{1}} }{{x}_{{n}} }\:,\:{n}=\mathrm{1},\:\mathrm{2},\:\mathrm{3},... \\ $$$$\mathrm{a}.\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{1}<{r}_{{n}} <\mathrm{2}\:\mathrm{for}\:{n}=\mathrm{2},\mathrm{3},\mathrm{4},\:... \\ $$$$\mathrm{b}.\:\mathrm{Diverge}\:\mathrm{or}\:\mathrm{converge}\:\mathrm{is}\:\mathrm{the}\:\mathrm{squence}? \\ $$

Question Number 31679    Answers: 2   Comments: 0

prove that sec^2 θ cosec^2 θ =sec^2 θ+cosec^2 θ

$$\mathrm{prove}\:\mathrm{that}\:\mathrm{sec}^{\mathrm{2}} \theta\:\mathrm{cosec}^{\mathrm{2}} \theta\:=\mathrm{sec}^{\mathrm{2}} \theta+\mathrm{cosec}^{\mathrm{2}} \theta \\ $$

Question Number 31677    Answers: 1   Comments: 0

24x^3 −26x^2 +9x−1=0(solve)

$$ \\ $$$$\mathrm{24}{x}^{\mathrm{3}} −\mathrm{26}{x}^{\mathrm{2}} +\mathrm{9}{x}−\mathrm{1}=\mathrm{0}\left({solve}\right) \\ $$

Question Number 31676    Answers: 0   Comments: 0

Given function f diferensiabel on R. If lim_(x→0) f((a/x)+b) non zero lim_(x→0) [f((a/x)+b)−(a/x)f′((a/x)+b)]=a how many value lim_(x→∞) f(x)

$$\mathrm{Given}\:\mathrm{function}\:{f}\:\mathrm{diferensiabel}\:\mathrm{on}\:\boldsymbol{\mathrm{R}}. \\ $$$$\mathrm{If}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left(\frac{{a}}{{x}}+{b}\right)\:\mathrm{non}\:\mathrm{zero} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[{f}\left(\frac{{a}}{{x}}+{b}\right)−\frac{{a}}{{x}}{f}'\left(\frac{{a}}{{x}}+{b}\right)\right]={a} \\ $$$$\mathrm{how}\:\mathrm{many}\:\mathrm{value}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{f}\left({x}\right) \\ $$

Question Number 31675    Answers: 0   Comments: 4

calculate lim_(x→∞) ∫_0 ^1 (x^n /(cos x))dx

$$\mathrm{calculate}\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\mathrm{cos}\:{x}}{dx} \\ $$

Question Number 31674    Answers: 0   Comments: 0

let p_n (x) polinom Maclaurin for function f(x)=e^x . How many degree minimal polinom (n) so ∣e^x −p_n (x)∣≤ 10^(−2) , for −1≤x≤1?

$$\mathrm{let}\:\mathrm{p}_{{n}} \left({x}\right)\:\mathrm{polinom}\:\mathrm{Maclaurin}\:\mathrm{for} \\ $$$$\mathrm{function}\:{f}\left({x}\right)={e}^{{x}} .\:\mathrm{How}\:\mathrm{many} \\ $$$$\mathrm{degree}\:\mathrm{minimal}\:\mathrm{polinom}\:\left({n}\right)\:\mathrm{so} \\ $$$$\mid{e}^{{x}} −\mathrm{p}_{{n}} \left({x}\right)\mid\leqslant\:\mathrm{10}^{−\mathrm{2}} ,\:\mathrm{for}\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{1}? \\ $$

Question Number 31673    Answers: 0   Comments: 0

let function−function f and g continues [ a, b] and diferensiabel (a, b). If f′(x)=g′(x)≠0, ∀x ∈ (a, b) and g(a)=a, g(b)=b, find value ∣f(b)−f(a)∣.

$$\mathrm{let}\:\mathrm{function}−\mathrm{function}\:{f}\:\mathrm{and}\:{g} \\ $$$$\mathrm{continues}\:\left[\:{a},\:{b}\right]\:\mathrm{and}\:\mathrm{diferensiabel} \\ $$$$\left({a},\:{b}\right).\:\mathrm{If}\:{f}'\left({x}\right)={g}'\left({x}\right)\neq\mathrm{0},\:\forall{x}\:\in\:\left({a},\:{b}\right) \\ $$$$\mathrm{and}\:{g}\left({a}\right)={a},\:{g}\left({b}\right)={b},\:\mathrm{find}\:\mathrm{value} \\ $$$$\mid{f}\left({b}\right)−{f}\left({a}\right)\mid. \\ $$

Question Number 31672    Answers: 0   Comments: 2

for what value p is a series Σ_(n=1) ^∞ ((1/n)−sin (1/n))^p convergens ?

$$\mathrm{for}\:\mathrm{what}\:\mathrm{value}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{series} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{{n}}−\mathrm{sin}\:\frac{\mathrm{1}}{{n}}\overset{{p}} {\right)}\:\:\mathrm{convergens}\:? \\ $$

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