Problem 1: For each of the following power series, find the the interval of convergence and the rational function whose Maclaurin series is equal to the power series on that interval. (a) f(r)=1 – x + a² – a³ + xª + • .. (b) f(x) = 1+ 2x + 3x² + 4x³ + 5xª + • · . (c) f(x) = 2x + 4x³ + 6x³ + 8a7 + · ..

Q: 1. Find the Fourier cosine series of the function f(x) given by if 0<x< 2; 4-x, if 2 ≤ x < 8 and…

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Q: Problem-5: Find the radius and interval of convergence of (4x – 12)" (-3)+2 n2 + 1 1. the power…

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Q: Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 5x + 5x² + 4x³ +... and _g(x) =…

A: Given power series  f( x ) = 2 + 5x + 5x2 + 4x3 + ......g( x ) = 2 + 9x + 2x2 + 4x3 + ......h( x ) =…

Q: Find the power series of f(x)g(x), given f and g as defined. f(x) = g(x) =  ∞     n = 1   x 3   n…

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Q: 1 1. Write the sum of the first four nonzero terms of the power series for - (Hint: 1- 2x 1 Use the…

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Q: 3. Let F(x) = sindt. cx (a) Show that F(x) = x- 3.3! x5 + by integrating an appropriate power…

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Q: 6. Find the Maclaurin seriés for f(r) = (1-x)-2. %3D

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Q: a. Show that if two power series E-oa„x" and E-o b,x" are convergent and equal for all values of x…

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Q: The interval of convergence of the series Σ(√n+1-√√n) (x+2)" is given by A=0 A-3≤x<-1. OB-3<x≤-1.…

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Q: 3. Let F(x) = sindt. (a) Show that F(x) = x −· 3.3! 5.5! 7.7! + by integrating an appropriate power…

A: The objective of this question is to express the function F(x) = sin(x) as a power series and then…

Q: Give the first 5 non-zero terms in the power series expansion and evaluate the derivative of the…

A: As per the guidelines I am supposed to solve one question at a time. Please repost other question…

Q: Σ 00 00 If f(2) = E n! 2" and g(x) = >(- 1) ", find the power series of (f(x) + g(x)). n=0 n=0 00…

A: we need to find 1/2(f(x)+g(x))

Q: f(x)=sinh(x)= n 2n+1 (2n+1)! =x+ + f +³ + +. 3! 5! 7!

A: We have to write the series for  f(x)=sinh(x)

Q: (b) Compute 95% confidence interval for o. Note: 20.025 = 1.96.

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Q: (a) Find the Fourier series for the extended function. (b) Sketch the graph of f(x) as well as the…

A: We are going to do part (c) We will find Fourier series and then sketch

Q: Consider the power series f(x)=log(n)x" . The R.O.C. of f(x) is (a) 0 (b) 1 (c) 3 (d) o

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Q: 3. Use power series centered at x = 0 to find the recursion formula and the solution for (x2 + 1)y"…

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Q: 1. Using one of the existing series in Table 11.5, obtain a Maclaurin series for f(x) 2. What is the…

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Q: 1. Given that f (x) = 2, find f(m) (0) using Maclaurin series.

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Q: Show that the series ∞ 1 Σ n² +1 n=1 converges using the Integral Test. Don't forget to state…

A: We know the series integral test for convergence, If fx is positive, decreasing and continuous…

Q: EX Use Taylor/Maclaucin's Series to.solve the following Ivps. (2) xycx)-xy)+ yo). -o. y(0)= 2 ýlo) -…

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Q: f(x)=In(1-x)

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Q: Find the interval of convergence for the following series. Σ(3) 2η

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Q: 7. Find the endpoints of the interval of convergence of the following power series: 10n xn ê n=1 N…

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Q: 1) Discuss an uniform convergent, continuous ,integral, derivative sinnx for a series E n?

A: Given the series as ∑sinnxn2 First, we will prove that the series converges for all real x, by the…

Q: 1. The function f(x) satisfies: f(0) = 0; f'(0) = -1; f'" (0) = 4; f"" (0) = 54, and f(4) (0) = 8.…

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Q: (c) If the series n=1 1 an is convergent, then sin(an + 5) is convergent. n² n=1 Final Answer This…

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Q: 3. Without examining the convergence, find a power series generated by 1 f(x) =-at c = 5

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