a) For which values of x does f(x) =E n=0 converge and for which values ofx does it diverge? b) When the sum converges, what does it converge to?

Q: 1. Let z, be the sequence of approximations to the root of f(x) = 0 and r, is a suitably close…

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Q: Q2 Prove that the scries 2+ x2 3+X2 4+x is Unitormly convergem? in any interval?

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Q: If E C, (x – 5)" converges at x = 11 and diverges at x = 15, what can you say about: (a) the…

A: Find the value of R and proceed. Find the center as clearly here it is 5 and proceed just as shown…

Q: 1 The improper integral -dx x²+2x+5 O None O diverges converges to 3n converges to converges to

A: Solution:To find the integral ∫1∞1x2+2x+5dxFormula:∫1x2+a2dx=1atan-1xa+C

Q: The series: (-1)" tann n=1 Da. converges absolutely Db. converges conditionally Dc. diverges 8.

A: Answer is following.

Q: determine the radius of convergence of the power series representation of the given function with…

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Q: The series E is (i) Diverge (ii) Converge

A: The given series is: ∑n=1∞1n45 Which is of the form ∑n=1∞1np And p-series converges if p>1 and…

Q: 4) Find a power series for f(x) = X + cos vx and and state its interval of convergence

A: The power series of cos x is: cos x=∑n=1∞(-1)nx2n(2n)! Replace x by x: cos x=∑n=0∞(-1)nx2n(2n)!…

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A: We have to show that the following sequence or series of functions converges pointwise but not…

Q: (b) Give an example to show that the product (fngn) may not converge uniformly.

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Q: (c) Show that fn (x) : x2+n? converges uniformly on [0, 0).

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Q: Find the interval of convergence of E(-1)". n 2n 36" n=0

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Q: Exercise 2. Determine whether the sequence of functions (fn)n converges uniformly on S. fn(x) = nx"…

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Q: 4. Let r be a positive number. Determine the value(s) of r, if any, such that converges. As part of…

A: Solution is in step 2

Q: 00 6n–1 Зп+2 1 Apply the nth term divergence test to conclude if E converges or diverges.

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Q: 0. The improper integral | x²+4x+8 -0- converges to 4 converges to 8. None converges to 4 diverges

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Q: Two sequences of functions fn and gn that converge uniformly but fn In does not converge uniformly.

A: See the attachment

Q: Determine whether the series of functions E fn converges uniformly: (-1)n+1 Vn+ cos(x) D = (-0, 00).…

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Q: 1 Determine by the direct comparison test if Ln=1 2n+1 converges or diverges .

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A: We have to show whether the series converges or diverges.

Q: CoS“ n In n а. n2 n³ n=1

A: Solution: a) ∑n=1∞cos2(n)n2+ln (n)n3

Q: Find the interval of convergence of E n(x – 8)" n=2

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Q: Determine whether r{(1+E)*"}.. Converges or diverges.

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Q: The series ∑Cn(x+9)^n converges at x=−4 and diverges at x=−17.

A: We know that if the power series ∑ n=1∞an (x-b)n is convergent at x=a, then this series is…

Q: (8) Find all k so that aek= dx converges. In this case, find the value.

A: Given problem:-

Q: B) Find the series' interval of convergence of E (tan-1x)" n%3D1

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Q: Find all real numbers p for which the improper integral ∫∞1t−pdt∫1∞t−pdt converges. For what values…

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Q: 5. Find the radius of convergence and the interval of convergence for E- (x – 2)" n%3D1

A: To find the interval and radius of convergence.

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A: Given:  So, the nth derivative is:

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A: The series is  . All the   are even and that's why consider .

Q: If E, 2"an converges and Eo(-1)"2"an diverges, then the radius of conver- gence of Eo anx" is 2. =0

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Q: 10.Does 00 sin? (n) 2n n=1 converge or diverge? Use any method and give reasons for your answers.

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Q: If E, 2"an converges and E o(-1)"2"an diverges, then the radius of conver- gence of o anx" is 2.

A: According to the given information, Suppose,

Q: n! Determine whether E- converges or diverges. Zn=0

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Q: 7. Let x₁ = a > 0 and Xn+1 = x + 1/x,, for n E N. Determine whether (x,) converges or diverges.

A: Q7 asked and answered.

Q: Determine the values of p for which f(x)da converges where f(x) = = 1-e-z TP

A: Here, in the question for what value of p the integral is convergent. The simple improper integral…

Q: 3. Find a power series representation for the function f(x) = = 1 x+2 and find its interval of…

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Q: 1 - dx converges? What are all values of p for which x2p 8.

A: Given : ∫1∞1x2pdx

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Q: Suppose that fn(x) and f(x) are functions R –→ R. Suppose also that fn converges uniformly to f as n…

A: Suppose that fnx and fx are functions from ℝ to ℝ such that fn converges uniformly to f as n→∞.  To…