4. Let a < b. Let x¡ = a, x2 = b, and Xn+1+Xn Xn+2 = for n > 1. 2 Follow these steps to show that (x„)n€N is Cauchy. (a) Draw a picture and let L = b – a. (b) Use induction to show that xn+1 – Xn] = L/2"-1 for each n. %3D

Q: 3. Find the Taylor series expansion of f(x) = xe* centered at xo fifth degree. 0, up to

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A: Consider the function  Compute Taylor series of the function about x=1

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A: The given function is f(x) = ex + sin(x) we have to find taylor series expansion about x = pi.

Q: 1. Prove that the Taylor series expansion for the function f(x) =x' about x 1 is reduced to x'.

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Q: Q/ Drive the Taylor series for: (X — 1). е* пеar X 3D 1 |

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Q: 7. (a) Find the Taylor series of the function f(x) = tan x at a = 4

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A: The given function is                         f(x)=e2x We have to find the Taylor series of f(x) at…

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Q: 41. Use the Taylor series generated by e* at x = a to show that (x – a)² e* = 1 + (x – a) + 2!

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Q: 4. Derive the Adams-Bashforth Three step method by the following method. Set Yti+1 = Yt; + ahf(t;,…

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Q: Expand the function f(2)=: in a Taylor series 1+z 1 centered at the point z, = -í

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Q: In the Taylor series expansion of exp(x) + sin (x) about the point x = T, the coefficient of (x- T)2…

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Q: = 3. Find the Taylor series expansion of f (x) = e* with a

A: Taylor series expansion:  Taylor series expansion of function f(x) about x=a is,…

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A: the Taylor series expansion of f(z)=1/2 is f(z)=1/2. the explanation is given below

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Q: 5. Find the Taylor expansion of f(x)=x² - 2x + 4 at x = 2.

A: To find the Taylor expansion of fx=x2-2x+4 at x=2.

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Q: 4. Derive the Adams-Bashforth Three step method by the following method. Set Yt,+1 = Yt, + ahf(ti,…

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A: Use Taylor series expansion

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Q: Q. In the Taylor series expansion of exp(x) + sin (x) about the point x = T, the coefficient of (x-…

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Q: Expand the function using Taylor series expansion. h(t) = 2t(1+t) in powers of (t – 1)

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A: The Taylor series expansion of the function f(x) about the point x=a is f(x) = f(a)+(x-a)f'(a) +…

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Q: 2. Find the Taylor series of f(x) = Inx at a = 1

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Q: Find the Taylor series for f centered at c = 3 if f(3) = 4 and f'(x) has a Taylor expansion (x – 3)"…

A:

Q: 3. Find the Taylor series expansion of f(x) = xe* centered at xo = fifth degree. 0, up to

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Q: sion of f(x) a

A: The Taylor series expansion of f(x) about x=a can be represented as,

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A: Solution

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Q: Q4: drive the Taylor series for (х + 2)е* for х %3D —2 X =

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A: This Problem is solved using concept of Taylor series theorem.