Q3. Use the Maclaurin expansion of er and the definitions sinh x = (e* – e-*)/2 and cosh x = (et + ex)/2 to x3 x5 show that sinh(x) = x + + + and cosh(x) = 1+ + x² x4 2! 4! 3! 5! + ...

Q: 6. Assume that the Fourier series of a function (x) on [-L, L] converges to f(x). (a) Derive the…

A: Given- Assume that the Fourier series of a function fx on -L, L converges to fx.  To find-  Derive…

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Q: 10. Let [ A= 103 -1 21 02 Calculate det (A) by cofactor expansion.

A:

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Q: Using the binomial expansion n (1 + x)² = ± (1) x², k=0 prove the following relations n n…

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Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

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Q: 0.2 Differentiate In x+1, with respect to x. x²-x+1

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Q: 2. (a) Use Taylor's theorem to find the first degree expansions of (i) sin x, (ii) e and (iii)…

A: for the function  f(x)  taylor polynomail of first degree about  x=0 is p(x)=f(0)+x f'(0)

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Q: 06) Expand in Fourier Series f(x) = 1x1 in -2<x<2

A: What is Fourier Series: A Fourier series is a trigonometric series representation of a periodic…

Q: Find the principal value of [in (1-j)]^ A) 4.71+2.08j B) -4.71-2.08j C) -4.71+2.08j D) 4.71-2.08j E)…

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Q: dy let y = (In z)" ln then da is Select one: a. (In r)* u (1 + In(In r)(1+ In r)| b. (In r)* n (1 +…

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Q: 7)Use Taylor theorem to estimate cos(0.23) with Jerror|< 10-2

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Q: Show that sin (2n + 1)z 2n +1 { 7/4 */4 0<まく

A: We will use complex analysis to solve the given infinite sum.

Q: Evaluate the following * S sin(4x) dx O cos(4x)/4+C -4cos(4x) + C O-cos(4x)/4 + C O-4cos(4x) + C

A: We can find the answer as below.

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Q: 2.Expand the following in powers of x i)√(1 + sinx) ii)e cosx iii) cos x²+

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A: Maclaurin series of f is given by, ∑k=0∞fk(0)k!xk =f(0) +f'(0)x + f''(0)2!x2 + . . . + fn(0)n!xn + .…

Q: (a) Determine the expansion of (1.003)*, correct to 4 decimal places.

A: Since you have posted multiple questions we will solve only first one for you. If you want any…

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Q: Determine the coefficient of the x'y term in the expansion of (2a + 3y) 2.

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Q: 6. What is the coefficient of x'y" in the expansion (4x + 5y) s. Show your work here:

A: We find coefficient of x7y8 in the expansion (4x+5y)15

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Q: a) Compute the quotient (22 + 2i) / (2 + 2i) b) Suppose z = (In 2) + i (pi/2). Compute e^z

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Q: 1.1 Find the interval of convergence for the binomial expansion of V(x? +2-x) (i.e. the values of x…

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Q: 3x Forf (x) = − ³ + 1, ƒ(− 10) = 5

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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),…

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Q: it is shown that the number of multiplication needed to compute x100 is 8. Using the same method,…

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