Let an bigger or equal 0 and sn = a1 + a2 + . . . + an (and s0 = 0). Prove that if an is divergent, then an/1+sn-1 is divergent.

Q: Find the Taylor's or Laurent's series expansion of the complex variable function which is…

A: Here we start the solution. Please check each step.

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: Let z = ƒ(u, y), u = ax + by, and y = ax - by. Express zx andzy in terms of fu , fy , and the…

A: Given, Let z=f(u,Y) u=ax+byY=ax-by

Q: If Osf(x)s g(x) on [a, ∞) and | g(x)dx is divergent, then | F(x)dx is divergent. Select one: O True…

A: We have to state true or false.

Q: (a) Suppose X U (0,n), where n is a positive integer X-[X] U (0, 1). (b) Use the result of (a) to…

A: De Moivre's theorem gives an axiom for computing the powers of complex numbers. We first gain some…

Q: (b) Let f(t) = (t- 2)*(t +1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: 59. Prove the general formula iwollo oi ons dx 1 x - In x – b + C %3D (x – a)(x – b) a – b - - where…

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: Q2= A) for E= 2xĭ-3Y3+K %3D prove That divc curl È)=0 %3D

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Q: Let f(t) = (t – 2)*(t+1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: If la,} and {b„} are divergent then {a, + b,} is divergent. True False

A: Given: an and bn are divergent . The objective is to state whether an+bn is divergent is true or…

Q: (b) State (without proof) the dimension formula for the linear transforma- tion L:V → W where both…

A: Dimension theorem

Q: Give V = C([0, 1]) the inner product defined as follows: (f, g) :| f(t)g(t) dt for all f, g e V.…

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Q: 2) Find the homogeneous linear ODE for which the given functions are Basis. -2x -2x , N, = Xe ", y3…

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Q: Let f(z) = ((z – 3i)² + 9)ez-si The Laurent series representation of f(z) in the domain 0 < |z – 3i|…

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Q: 1 prove that | e -ym dy = mr(m) -ут

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Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: -пя for all x € 0, 00). t fal(x) — же па for all z € [0, co). Find || fn||sup for all n. Prove that…

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Q: * Let A E R"Xn, ĩ E R" \ {0} and A = 0. Prove that if the origin is stable if and only if ī is…

A: Given A∈ℝn×n,x¯∈ℝn\0 and Ax¯=0 To prove that the origin is stable if and only if x¯ is stable.  If…

Q: and let P(x) =E sin kx its partial sums. Show that k? Let f(x)= 5sin kx k=1 k=1 E,(f) <!. n

A: Given, fx=∑k=1∞sinkk2 and Pnx=∑k=1nsinkxk2 Let, fkx=sinkxk2 Since,fkx=sinkxk2≤1k2

Q: If z = f (u) + g(v), where u = = and v = x ·y. Prove that Q1: x· Zx – y· Zy = y² · Zyy – x² · Zxx…

A: We have z=fu+gv, and u=xy  and v=x.y We need to use the chain rule and prove x.zx-y.zy=y2.zyy-x2.zxx…

Q: Use the generalized chain rule Question of Let f(x, y, z) = 2x(y+ z), x = uv, y = v + w, and z = w².…

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Q: prove that F(r) = 4z-3z is chaotic on the interval [-1,1]. (Hint: consider g(6) = 30 on S'.)

A: The given function: F(x)=4x3-3x The given interval is [-1,1] To prove that F(x) is chaotic: Chaotic…

Q: Let T : (0 → l defined by T(Xn)) = ( n. (a) Prove that TE B(l, l∞) and ||T|| = 1.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let (An) and (Bn) be (Fn)-martingales on (2, F, P) and let Cn = A - Bn. Show that E [(Am - An) |Fn]…

A: Solution:

Q: If z = f(u) + g(u), where u = = and v = x · y. Prove that x · Zx – y · Zy = y² ·Zyy – x².Zxx using…

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Q: Q3: a) Let ƒ : Z → Z be such that f(x) = Vx² – x + 1. If invertible, and if it is, what is its

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Q: Let A and B be constants. Prove that ifA &gt; 0, then all solutions of dy/dt+ Ay = B approach the…

A: On some rearrangement you will get,

Q: the potential function f for the field F = 5x i + 9y j + 4z k. eral expression for the infinitely…

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Q: 4. Evaluate the Kullback-Leibler divergence between two Gaussians p(x) = N(xu, E) and %3D q(x) =…

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Q: If f(x, y, z) = In(x7 + yº + z®) + (x° + yº + z®), then find fr, fy and fz at (1, 1, 1).

A: →fx=∂f∂x, During derivative of ∂f∂x, consider 'y' and 'z' as a constant→fy=∂f∂y, During derivative…

Q: If /has an isolated singularity at z a and /(2)- an(z-a)" is its Laurent's expansion about z a. Then…

A: We are provided that f has isolated singularity at z=a. The Laurent's expansion about…

Q: (1) Let F(x, y, z) = (1, 2yz², 3z² + 2y²z). (a) Find the divergence of F (b) Show that is…

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Q: 2n Jnsi (x) = " J,(x) – J(4)

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Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

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Q: The signal (t) = tan(t) satisfies the Dirichlet conditions. O True O False

A: False

Q: Find the divergence of F = (1 + xy)i - y°z?j - xz°k | Select one: 2 2 O a. x-3y z-3xz? а. х-Зу z…

A: We have to find the divergence of F=(1+xy)i^-y3z2j^-xz3k^ .

Q: Let A € R"Xm, n > m. a) Use the SVD of A to deduce the SVD of AT A. b) If m = n and A is full-rank,…

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Q: let a t4° フ+一 5 an +4nwhich of the follewing slatements colrecd ? are I) { cn? is divergent…

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Q: Defn `x;>-` "x; diverges to minus infinity" VR>O such that Vj >Ng *;<-R NR HWs If x→0 then → O.…

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Q: Let f (x) = e* cos x. Use the second-degree (n 2) Taylor polynomial P2(0.5) about x, = 0 to…

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Q: Defn `x→- `xj diverges to minus infinity" VR>O 3N, such that Vj zN *j<-R 8;<- R Hw7 het x; = - 8j…

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Q: Find the Taylor's or Laurent's series expansion of the complex variable function which is…

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Q: If F G) and g(x) are conbinuous on [anb] andit =Sprove thab. SF= 58 prove thab Icefanb] Such that…

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Q: QV A/ Find at (t-) as a funetion of t, if (x = t).(y= vE)

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Q: Prove that the Generating function of Bessel function i.e. £1J„(x)t" = e*/2(-1/2), %3D

A: Let ϕx,t=ex2t-1t. Let us prove that ϕx,t is the generating function of the Bessel function…

Q: Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the…

A: Here we have to use Euler's method to obtain a four-decimal approximation of the indicated value.…

Q: 5. Compute the 1st and 2nd degree Taylor polynomials L(r, y) and Q(r, y) for f(r, y) = In(r² + y?)…

A: 5 Given function is fx,y=lnx2+y2. We have to find the 1st and 2nd degree Taylor polynomial Lx,y and…