b) Let S = (/n .ne N}Uu(1+3/2n •n € N}u {6 – /3n•n € N}. Find S' and sups.

Q: Let S, := (1 - (-1)"/n :n e N). Find inf S, and sup S,.

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Q: let f: [o,0o] - [o,00) be decined by Fla) =x? IF is a homeomor phism Detemine

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Q: Show that cl S = SU(Bd S) %3D

A: LetS⊆R, claim: closure(S)=S∪Bd S =S ∪ S' by definition bd(S) ={ x∈ℝ/x∉S° & x∉(ℝ-S)°} let  x∈S∪S'…

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: 30) Let n 2 3 and k e Zn. Prove that in Dn. p u = up"-k, %3D

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Q: Prove that Ks.s has (s-1)!s! s S13! . Hamiltonian walks for s > 1. 2

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Q: 2.16 If S = {r |0<< 12}, M = {x | 1 < z < 9}, and N = {r | 0<r<5}, find (a) MUN; (b) MON; (c) M'nN'.…

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Q: - Let X1, X2, ...,Xn~N(0,0) and T = Fi (T),Var(T)

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Q: Exercise 4 Show that for cach k EN it holds that kn-k < n! S n", Vn E N, n2 k. Deduce from this that…

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Q: Determine u(x,t) para o problema abaixo: a2u 25 ax2 u(0, t) = u(16, t) = 0, t>0 u(x, 0) = 0, du(x,…

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Q: 4. If x = cose + isin0 & 1 + V1– a² = na then prove that 1+ acose =(1+ nx)(1+) %3D 2n

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Q: Prove that T2 is chaotic on 1.

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Q: (2) O show that you Y2,43, and Yu are solutians to NOTE : Geueral Solufion

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Q: Let E Os 0s n, 0s zs 1, 8Vzs rS 8. Compute

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Q: 6. Given: 21 and 22 are complementary Prove: SX 1 WX S. 2,

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Q: Evaluate S, 5y²dV where E is the reqion below x + 3y + 6z = 9 in E the first octant.

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Q: Prove that Z(Sn) = {e} for n ≥ 3.

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Q: Let Sn=Vn. Find an and show that an O but that Sn=ER=1@k diverges. ->

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Q: in(Zg++s)indeyclic grauf2enerated by I, Z,3

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Q: Q4 Prove that E(",") = ("::)- () .n 2mzr. %3D +1 +1

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Q: Let X(t) be a WSS Gaussian random process with ux(t) = : 1 and Rx(7) = 1 + 4 cos(7). Find

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Q: Find an explicit formula for s, if so = 1 and %3D Sn =2+2s0 + 2s1 +... +2s, 2+ Sn-1, Vn >1.

A: Generating Functions for a sequence a0, a1,a2,………an,……………… is an infinite series of the form ,…

Q: Prove that the random process {X(t)} with constant mean is mean-ergodic, T T 1 if | C(t,, t2) dt dt,…

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Q: Find e² in the form of u + jv and le²| for z = 2 + 3nj

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Q: Derive the moment generating function of X ~ Bi(n, p) and using the MGF compute E[X] and Var[X].

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Q: S 1.Prove that L (coswt) = %3D s2+w2

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Q: O Find the set S on wlc fu x) = x Sin X convergas pointwise Cony

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Q: 4. Exercise §3.4 #47. Let n €Z with n> 0. Prove that E - a(n)

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Q: 1. Prove that L(t)=. 1 s2

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Q: 1. Prove that L(t)=. s2

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Q: Determine Z- Trans furm in cludig vegiva of con vergence in n 7o メ(At n<o

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Q: and p(n) iF IF Fon) = 4 sec t dz.

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Q: 6-1718 ss (a) For t= 1.735, should you reject or fail to reject the null hypothesis? prary O A.…

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Q: FIND THE UNTQUE SOLUTION OF AU= 0 34/x1<4 SUCH THAT U(x) 5 on x =3, U(x)=6 on /×/ =4 on 2 in R in R…

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Q: 1 Brandt r at an ans monthly within a • Deter order b Find

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Q: Then, we assume that Wn+1 - H Wn - u kin+1 kn 1 2 E Wn-p E Wn-p p=0 p=1 (6) Zn+1 - € Zn - E Vn+1 Vn…

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Q: 03) find the ax en the sl abetwed wol y=J2-X, and y= 0

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Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: If {X(t)} is a Poisson process, prove that n - r S P{X(s) = r/X(t) = n} = nC, where s<t %3D %3D

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Q: To test this null hypothesis, we compute t using a special case of the formula. state the formula?

A: In this case, we need to write the test statistic formula for testing the null hypothesis.