(a) Use induction to prove that for all x > 0 and all positive integers n x² + 2! e* > 1+x + + 3! n! Hint: Observe e = 1+ | e'dt > 1+ | 1dt > 1+x, and then [ e'dt > 1+ 72 | e'dt | (1+t)dt > 1+x + e = 1+ 2 (b) Use part(a) to show if n is a positive integer, then et > x" for all æ sufficiently large. Hint: this is equivalent to showing >1 for x sufficiently large.

Q: 8. Theorem 1.7.4 x -le* dx is convergent if and only if n > 0.

A:

Q: xdy + (3xy + y - e-³x) dx = 0

A:

Q: 5. Residue theorem Use the residue theorem to compute (a) ro∞ t2+1 dt.

A: Given integral is∫0∞t2+1t4+1dtThe integral is computed using Residue theorem as shown below.

Q: Evaluate z' if z = [1 + (x² − 1)³] ²³ O a. 2(2²-1)² (1+(2²_1)²³) i O b. none of these О с 41(2²-1)²…

A:

Q: Which of the following expressions' inverse Laplace transform is the solution to the initial value…

A:

Q: Find the Laplace transform of f(t) = 2 Bt + C 0 ≤ t < (D+1) t≥ (D+1)

A:

Q: Prove or disprove the following claim using a proof by induction. Claim: Vn € Z>1, [n even ^ (−1)n =…

A: We have the claim ∀n∈ℤ≥1, n even∧-1n=1∨n odd∧-1n=-1 This can be written as: for all integers greater…

Q: -. Let f(x):= and let F(x) be the odd periodic extension of f(x). (a) Plot F(x) on [-6, 6] (b) Find…

A:

Q: Let X - Poisson(a). Show that E X(X - 1)(X- 2).… (X – k)] = ak+1

A: We are given that: X~Poissonαfx=e-ααxx!         x=0, 1, 2,....

Q: Find the Laplace transform for -4 f(t) = e'sin2t + e¯'cost 2 s + 1 2 s + 1 + s + 2s + 5 s - 2s + 2 2…

A:

Q: Let f(x) = ex On [-∞, ∞] Part A) Show that f(x) dx Converges - Part B,) Find the Fourier integral…

A:

Q: Example 7-18. Show that eitx =1+ (eit - 1) x(1) + (eit – 1)². T+.. x(2) + (eit – 1)* . r! where xr)…

A:

Q: 3 Given the function f(x) = x + 3 on the interval 0 ≤ x ≤ 1 the Fourier sine series expansion of…

A:

Q: Use mathematical induction to prove that the n-th derivative of (x +n + 1)e" is given by d" (x+1)e =…

A: We use mathematical induction. The detailed solution is typesetted and given in detailed as the next…

Q: 3: You are given that the Fourier series expansion of f(x) = x2, x € (-7, 7) with period 27 is +4…

A:

Q: Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a. 2r π d) f(x) = sinx, a = 7…

A: The given functions are written below.(a) .(b) .We know that the order Taylor polynomial of a…

Q: X1,, Xn ~ N(-3,2) (random variables) ... Z, = 1(X,X2 + X1X3 + X1X4 + •..+ X1X„), n > 3 %3D (a) Find…

A: From the given information, X1, .....,Xn~N-3,2 And Zn=1nX1X2+X1X3+...+X1Xn, n≥3 Consider,…

Q: 2.9 Present an inverse-transform algorithm for generating a random variable from the…

A: Here is the inverse-transform algorithm for generating a random variable from the piecewise-constant…

Q: Let f(x) = 1 - 2x where 1 0 ≤ x ≤ Expand f(x) to a 4 Fourier sine series converging to 1 f(x) on the…

A:

Q: Prove that Q_n = sup 1/n(1-(1/(nx+1))=1/n

A: Answer

Q: Integral Representation r(c) Г(В)Г(c-b) Jo -2 F₁ (a, b; c; x) = for F, : ¹ (1 - ₁)¤-b-¹ (1-x₁)¯ª di…

A:

Q: u)n(p+u°О

A:

Q: Use a cosine Fourier expansion, as u(x,1) = u,(0) +E, (1) cosnx, to solve =e' cos2x in (0,1)×(0,+0),…

A: Given ux,t=u0t+∑n≥1untcosnπx∂u∂t-∂2u∂x2=etcos2πx  in 0,1×0,+∞∂u∂x0,t=∂u∂x1,t=0, t&gt;0ux,t=x,…

Q: The Laplace Transform of d(t) = * %3D 1/s^2 -1/(s+5) O 1/s O 1/(s-6) O 1/s^3 1-/s O t O 1

A: This question is based on finding Laplace transform of delta t.

Q: What is the kernel of Laplace Transform? O e^(-jwt) O e^(jwt) e^(-st). e^(st) Which of the following…

A: There are multiple of questiona, one from laplace transform,secont from real analysis and third from…

Q: H(n) = 2H(n − 1) + 1, n ≥ 2, H(1) = 1 ⇒ H(n) = 2" – 1↓ Using mathematical induction, verify that our…

A: We prove result in following way. Base Case: Result is true for n=2. Hypothesis : Assume result is…

Q: Compute the Laplace transform of (1) = e-³ u(x-4)

A: Consider the given function,The transformation rule is defined as,And,Now, find the value of .Use…

Q: 2) For the Fourier series of the function f, ao + Σo acos (kx) + bxsin (kx) we have |bk| ≤ 211/11.…

A: Solution:2)b)FalseWe can solve it by taking a simple exampleFourier series of any function f is a0+…

Q: Proof using induction method: a+ar+ar+ar'+... . +ar"=(ar" 1-a)/(r-1)

A: To prove the formula:

Q: expanded as odulus of x, the 1-e

A:

Q: Common monomial factor 7w^2-16w

A:

Q: A sequence is defined by the relationship r[n] == Ỹ h{m}h[n + m} = h{n} » h{_n], ma-00 where h[n] is…

A: As per the question we are given a discrete output sequence as : r[n] = (4/3)(1/2)n u[n] + (4/3)2n…

Q: find Fourier series [o, 2w] an <x<π fcx) = TくXく 2w

A:

Q: B2. Euler's Theorem Let G = U(n) %3D Prove For any a in G, a^m = 1 (mod n), where m = (phi) (n),…

A:

Q: Let xfn] be a discrete-time signal whose Fourier transform between -r and r is 4 |w< 4 X(e) = 0, 4…

A: Given Discrete time signal xn whose Fourier transform between -π and π isXejw=4πw2,  w&lt;π40,…