An ancient technique for extracting the square root of an integer N > 1 going back to the Babylonions involves using the iteration process N 1 Xn +- 2 In+1 = In where ro is the largest integer for which x2 < N. Show that if by change x² the sequence is constant, i.e., x„ = /N for all n = simply Newton's method applied to N, 1, 2, ... Show that the sequence is %3D f(x) = x² – N 1 How many iterations does it take to calculate v411 to three decimal places using xo = 20?

Q: Use Theorem 9.17 to give an upper bound to the 40th truncation error for the series (-1)" n2 n=1…

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Q: Determine whether the series 5 + 2 sin(3n) 9" + vn n=1 converges or diverges. 8.

A: See the attachment

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A: Given series, ∑n=0∞42n8nx+6n

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A: For the given series

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A: Consider the integral: I=∫012ex-1xdx……1

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Q: 7 A = > B=E(-1}**1 n2 n=1 N + 4n+1> n=1 ,3

A: The series A is convergent. The series B is divergent.

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A: We know by Ratio Test, the series ∑an is convergent if limn→∞an+1an &lt; 1.

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Q: 1 f(x) = 1– 4x2

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Q: Suppose that the power series Сn (2а — 3)" n=0 converges when x = 5.5 but not when x = -5.5. If its…

A: see below the answer

Q: 42n (x+6)" is convergent if The given power series : 8 |(z +6)| < (Write your answer up to 5 digits…

A: We have to find the convergent point

Q: Compute the limit of the nth term and use it to explain why the series is divergent. +00 2" – 1 2 2n…

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Q: 5. Use Theorem 5.10 (Section 5.3 in Vol. 2 of OpenStax Calculus) for this problem. How many terms of…

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Q: Σ 42n - (x + 3)" is convergent if The given power series : 9n n=0 |(x+ 3)| < ( Write your answer up…

A: using the fact that power series is convergent whenever |x-a|&lt;R;  R is radius of convergence

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A: If there exists a non negative real number 'r' (I.e., r ≥ 0) such that,  (i) the power series…

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A: The alternating series to be tested is ∑n=0∞(-1)n2(n+1)2.  Here, the nth term is an=(-1)n2(n+1)2 and…

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A: Given: The series  Here the kth term is

Q: Show that the series 71 n' (2n)! is convergent. Deduce that lim 0% (2n)! n=1

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Q: 8. Use the fact that ()1 converges to 0 along with the theory derived in this section and worksheet…

A: We can solve this by the help of squeeze theorem and given details

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Q: 1. f (x) = 1+4x2

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Q: Use any method to determine if the series (Зп)! (n!)³ n=1 converges or diverges. Show your work.

A: To find the Converges or Diverges of the given series.

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Q: Suppose that Cnxn converges when x = -4 and diverges when x = 6. What can be said about the…

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Q: If a power series in (x – 4) converges when x = 11, then it must also converge when x = -2. |

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.