please explain with clear steps why is it important to center the  power series at the same x value as you have in your initial conditions?

Q: Find a series equivalent to f x² sin(r?) dx and then approximate its value using its first four…

A: According to the given information, it is required to find the series for the given integral.

Q: fin the values of p for which the series converges. The series is: the sum from n=1 to infinity of…

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Q: Find a power series for the function 3 f(z) = 5- 2r and determine its interval of convergence.

A: from geometric series,

Q: The value of r can be calculated using the following series: 1 n = 4 ) (-1)"-1 2n – 1 Add terms one…

A: Given series is π=4∑n=1∞(-1)n-112n-1 expanding we get…

Q: If the series En=1 an converges absolutely, the sequence {b,} is bounded, then the series sin(a,)b,…

A: ∑ann=1∞ is absolutely convergent and bn is bounded .

Q: : For what values of (x) does the power series converges? Find the Serie interval of convergence?…

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Q: 1. Find the power series E-o anx" of f(x) = e-x²/2. %3D

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Approximate the sum of the series correct to four decimal places. (−1)n + 1 n² 0.0158 n = 1 X

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Q: Test the series for convergence or divergence. (-1)^-1 √n-5 n = 26

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Q: 23: Test convergence for th 1- En-o)(x + 5)"

A: According to the guideline I am answering only first question. 1- ∑n=0∞ 34nx+5n   Ratio Test : A…

Q: 1 f(x) = 1– 4x2

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Q: Use substitution of power series to approximate cosh (2x) dx

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Q: Problem 6. We've solved the equation y"(x) + y(x) = 0 in class by assuming the solution is in the…

A: We solve the Differential equation  y"+y=0.....(1) about x=5.

Q: Show that the series Σ (-1)"+1 log(n + 1) converges.

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Q: If f(x) = 2x" and g(x) Σ na", find the power series of f(r)g(x). %3D n=0 00

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Q: The function f(x) = ekx² can be expressed in power series form. If A is the approximate value of…

A: fx=ekx2k=3

Q: 1. Find a power series representation for the function f(x) = 5x2- Determine the interval of…

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Q: Verify F (1,1; 2;x) = –x¬ln(1 – x) By expanding each function in a power series about x = 0.

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Q: Find first two terms of the Taylor 3 d) ( series centered at a = 2 of the function f(x) x+2°

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Q: Find the sum of the convergent series. Στο(-3)"

A: Given the seriesThe series is a geometric series with common ratio r=−56 and the first term a=1.The…

Q: Write the first two nonzero terms of the Maclaurin series for f(x) = arcsin x. Justify your work.

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Q: 4) (a) Find a deqree 3 power series for the function sin(x) centered at T fla) = f (2) + f(a) (x-a)…

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Q: 2 + In n Use the Comparison Test to determine if the series E converges or 2 n=1 _n sin'n diverges.

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Q: 5(x-3)" 00 Σ find the interval of convergence For the power series: n² n=1

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Q: We the first three terms of the binomial series to approximate f(x) = 1 at x = 1/2. Then find f(¹/₂)…

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Q: Estimate the series Σ n=1 (-1)" to within an accuracy of 0.01

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