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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