4. (a) Test the series n=1 for convergence and divergence. (-1)^ (1+x)" 2n n² Hence, deduce that + 32 52 f(x) = (b) Find the Fourier sine series of the function x₂ 0

Q: 1 (a) Use the Integral Test to show that > converges. 2. n' +1 d 1 n = 0 -1 tan dx Hint: 1+x?

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Q: A 4 cm Q R. В ai All In the diagram, the length of the arc AB is- 10 times that of OA. Given that OP…

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Q: 2. Using the attached table of Power Functions, find the power function for the given functions and…

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Q: (-1)*+12"(a – 1)² n · In(3) n=1 When a=0.5, (-1)*+12"(0.5 – 1)" n· In(3) n=1 Using p-series test,…

A: The given series is ∑n=1∞-1n+12na-1nn.ln3 Here a = 0.5 is given. Hence the series becomes,…

Q: 3. Find the derivative of the series (-1)**'2" -x2n+3 n=0 n!

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Q: Use the power series =o(– 1)"x" to determine a power series centered at 0 for the function f(x) =-…

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Q: sinx x0 e²x-1 5. Use power series to evaluate lim

A: Power series expansion

Q: e graph of the

A: Here, in part b we have to sketch the graph of the function to which fourier sine series converges…

Q: Find a power series for the function centered at 0. 3+x? O(-1)"x" Σ n=0 3" n=0 3 +1 5(-1)"x" +1 n=0…

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Q: (iv) COS x -dx, 3.

A: We will answer the first question only as we don't answer multiple questions at a time. Please…

Q: 4. Obtain (a) the half-range cosine series and f(t)= (b) the half-range sine series for the function…

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Q: 24. Let f be a function such that f'(x) Maclaurin series for f? = sin 9 (A) x - + 10 216 9 X (B) 2x…

A: Step 1:Step 2:

Q: Σ (-1)". It is known that for all x E R, COS x = (2n)! n=0 (a) Find a Power Series Representation…

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Q: (a) Write out the power series for 1 f(2) = zexp(iz?) + 1+ 2iz at zero.

A: The function is fz=z expiz2+11+2iz. As the power series is ex=1+x+x22!+x33!+…+xnn!+… at a = 0, then…

Q: Q3-A) Solve the following equation: 3y² cos 2x dy=y' sin 2x dx B) Find the radius and the…

A: We need to solve equation.

Q: (a) Find the Taylor's series for the function f(x)= In(3+4x) at x= 2.

A: The Tayler's formula can be used to calculate the expansion of the function. The formula for the…

Q: (a) Find the Taylor's series for the function f(x)= sin x at x = (b) Using L'Hospital 'rule to find…

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Q: X Expand f(x) = sin 0 < x < π, in a Fourier cosine 9 series. In addition, indicate when the Fourier…

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Q: 5. Let f(x) = -1 0<x<T 0 π < x < 2π 1 2π <x<3π (a) Find the sin series for f(r) with period 67. (b)…

A: 5) (a)

Q: 2 Obtain (a) the half-range cosine series and (b) the half-range sine series for the function 0. 0(…

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Q: Q 3. Find the Taylor series of f at the point c for : (a) f(x) = e", c= 2 (c) f(x) = sin x, c= T (b)…

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Q: 6. Evaluate the indefinite integral as an infinite series. (b) / sin z (a) / r cos(r)dr

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Q: Q6. (1) Given log(1+t) = t = t + t² 4 valid for 1411, determine the power. series for log (1+2x) up…

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Q: 3. Consider function f(x) given by the series 8∞ f(1) Σ- 2π)" n=0 Find a power series for f'(x) and…

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Q: 6. (a) Find the Taylor series for f(x) = e-2* centered at a = 3. For what x-values is f(x) equal to…

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Q: 4. Find the function given by the power series n f(x) = Σ + ²)", x+2 when -7 < x < 3. 5 n=1/

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