(iii) 2 an must converge if lim an 0. n =0 bn (iv) 2 bn must diverge if b, > 0 and lim = 4. n->0 (1/Vn) n=1

Q: x2n-1 Find the radius of convergence of En=1 5n 100

A: Given, ∑n=1∞ x2n-15nFind radius of convergence

Q: 8. Theorem 1.7.4 x -le* dx is convergent if and only if n > 0.

A:

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A:

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A: It can be obtained by using limit evaluation of the given sequence

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A:

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A: Solution is given below

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Q: QDTest for convergent and divergent : (show your work) e-2 1. ΣΟ-1 n+cos²n

A: answer : this is divergent

Q: 17. Evaluate lim x-0 1+x-ex 4x² using Taylor series. You can use that e 8 -Σ² k! k=0

A: I have given the answer in the next step. Hope you understand that

Q: Which of the followings are false? 1 1. En-1 sin-| converges by comparing with 2 II. En-1 converges…

A:

Q: 5a) When p is a real number greater than 1, it can be shown that the improper 1 1 integral |dx…

A: Convergence of Improper Integrals at left-end point: Let us suppose that x=a be the point of…

Q: +00 Inn E(-1)" n2 n=2 divergent convergent

A: To solve the following problem

Q: Question 6. In Example 2.4.3, show that f(r) converges to 0 at each r E [0, 1].

A: Let us consider the function :                                  fnx=n if 0<x⩽1n0 otherwise We…

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A: We have to check

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A:

Q: (x + 2)" Determine the interval of convergence of 3n +1 n=1

A:

Q: 8 (iv) [(-1) n=1 n :)" sin n n+1 [ Select] [Select] is conditionally convergent is divergent. is…

A:

Q: • If it converges, what is its limit? Provide some justification for your answer. • If it diverges,…

A: key points By monotone convergence theorem ;  if a sequence is decreasing and is bounded below by an…

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A: Follow next step

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A: Given that,

Q: 1.1. For r € (0, 1] (1) = f(r), wher

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Q: Example 2.3.16 Determine the following integrals are convergent or divergent 8 for e-2x dx

A:

Q: Σ3" 6.6 Find the function represented by the power series 3nt > E". Determine its interval of…

A:

Q: ii. Investigate convergence or divergence of 2n + 100 3n +1). n=2

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A:

Q: Problem 7. Suppose that xn 2 0 for all n E N and that lim ((-1)"xn) exists. Show that (xn)…

A: Let lim -1nxn=1 Therefore, For ε>0 ∃ k∈N such that -1nxn-l<ε2 ∀ n≥k ⇒xn+l<ε2 ∀ n≥k, and n…

Q: 4. Let r be a positive number. Determine the value(s) of r, if any, such that converges. As n" n=1…

A: Explanation of the answer is as follows

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A:

Q: CoS“ n In n а. n2 n³ n=1

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

Q: ∑∞ ((−1)^n−1(1−n)) /(3n−n^2) n=1 Find if the series converges or diverges! Find the limit if it…

A: Topic- Series

Q: 1. Find the interval and radius of convergence 00 (4x – 2)" (-1)" (4n + 1) n=0

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Q: n = 1 Vn lim n-00 converges O diverges

A:

Q: - When Using Newton iteration to estimate the roots of the equation f(x) = 0. Find the order of…

A: Given:-

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A: Given series is ∑n=0∞-x3n8n(2n+1)

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Q: .2 A limit by Taylor series Evaluate lim x + 2 cos x – 2 3x

A: Given: limx→0x2+2cosx-23x4 we know that cosx=1-x22!+x44!-x66!+.......................…

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A: I proved step by step two implications

Q: (3x +1)" +1 ? 2n +2 What is the interval of convergence of the series Ln =1 2 b. -<x<0 Oc. sx<0 2 d.…

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Q: 2(2n) 3" (x+1)" n=00 Find the radius and interval of convergence for series Σ 12n n=1

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Q: -5 1 10) Determine whether the improper integral diverges or converges: ī dx Vx-1

A:

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Q: 1 -dx x²+2x+5 -00 3n converges to O diverges converges to 3 4 O None converges to 4 converges to

A: Answer is 3π/8.   First option is correct one.

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Q: 23. Suppose N is a positive integer, a, -f(n), and f(x) dx diverges. What does the integral test…

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Q: Use series to evaluate the limit. 1- cos(5x) lim x → 0 1 + 5x – e5x

A: Authoring guidelines: As per the authoring guidelines we can solve only the first question.Please…