Problem 11.3.8. Observe that for all x = [−1, 1] |x| ≤ 1. Identify which of the following series converges pointwise and which converges uniformly on the interval [-1,1]. In every case identify the limit function. (a) Σ (x¹ - xn-¹) n=1 (b) (x²-x²-1) n n=1 (c) En=1 (x²-xn-1) n²

Q: #3. Prove that tiz 2 = tan [ + log (+ +12) ³ ] iz #4. Find the Laurent series for the function 1…

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Q: How would I find if the sequences an=nπ cos(nπ) and an= (n2)(1/n) are convergent/divergent?

A: To examine the given sequence regarding convergence, that is study the limits as n tends to infinity

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: Approximate the series of e with the finite sum M (−1)n SM(x) = Σ -(x-15)²n. 2nn! n=0 Use a graphing…

A: fx=e-x-1522 and the approximated sum function is given by SMx=∑n=0M-1n2nn!x-152n

Q: +∞ If f(x) = (x-2)¹" √n-1 O O O 1-2 n=2 V +00 Σ(-1) ¹(2-2)-1 n=2 √n-1 n(x-2) ¹√n-1 +00 -n(2-2)√√n-1…

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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: Σ (-1)"x2n+1 ren: tan' for all æ E [-1, 1] 2n + 1 n=0 2x Find the power series representation of…

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Q: Find a power series representation for the function = x sin(5x) on (-∞, ∞).

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Q: #8: If the sum s of the series 3(-1)" (n+2) n=0 is approximated by 3(-1)" Sm (n+2) find the smallest…

A: The smallest possible m will be , m = 2.

Q: Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 5x+3x²+2x³ + and g(x) = 6 + 3x +…

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Q: 2n ƒ(~) = 2 – (* (x - 5)" To determine [.. 4.9 ¶¸ ƒ(2) f(x) de to within 0.0001, it will be…

A: The Taylor series approximation is very significant in expanding a function in a power series of x…

Q: 1. 2. Find the interval of convergence of the series (-2)(x−1)ª 4* (k+1) k=l -x² Evaluate ¹* (x² +…

A: Using the results: Radius of convergence, Leibniz alternating series test, p-series diverge for p=1

Q: 6. Consider xn+1 = (1/3)(2xn - 9/xn2). Does it converge for any nonzero initial point? If so, to…

A: Given that,

Q: Suppose that f(x) and g(x) are given by the power series f(x) = 7+ 4x + 3x² + 3x³ +... and g(x) =…

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Q: 3. f(x) = Expand the 2π-periodic function π, 0 < x < T 2π-x, π < x < 2π in a Fourier series. In…

A: Here the function is given by fx=π,            0&lt;x&lt;π2π-x,      π&lt;x&lt;2π The 2π- periodic…

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: Suppose that f(x) and g(x) are given by the power series f(x) = 3 + 4x + 3x² + 2x³+... and g(x) = 6…

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Q: Suppose that T(0) = a and T(1) = b are some constants. Define the running pairwise average as, for n…

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Q: Suppose that f(x) and g(x) are given by the power series and CO = By multiplying power series, find…

A: f(x)=5+5x+2x2+2x3+.......g(x)=4+2x+5x2+4x3h(x)=f(x).g(x)

Q: 3. Prove that the sequence of functions (fn)1 converges uniformly on X CR if and only if the series…

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Q: We define a new function h via the equa Notice that h is defined as a power series this.) 1. Find…

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Q: Suppose that f(x) and g(x) are given by the power series f(x) = 4 + 2x + 3z? + 2a + %3D ... and g(a)…

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Q: Suppose that f(x) and g(x) are given by the power series f(x) = 5+ 2x + 6x? + 3x³ +... and g(x) = 2+…

A: Series.

Q: For the following function, find the full power series centered at x = 0 and then give the first 5…

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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: 4. Using power series, one can show log(x) = (x − 1) + O((x − 1)²) as x → 1. Use the definition of…

A: Big-O notation: Let f(x) and g(x) be two real valued functions .if there exists a constant C and a…

Q: (In n)ª Problem 2. Use the Alternating Series Test to show that for any real number a, the series…

A: The given series  is : ∑n=1∞-1n+1ln nan  Applying Alternative series test to the above series is…

Q: Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 6x + 4x² + 3x³ + ... and g(x) =…

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