9. By integrating term by term from 0 to x a power series representation for In (1-t), show +00 that >. x" = x+ (1-x) In (1- x). (п- 1) п n=2

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A: To find the power series representation centered at 0 of ln(1 – 3x) and its interval of convergence.

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Q: 2. Ln=0 (n+1)* If the series converges, use the Integral Test can be used to show convergence

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Q: F(-1)"(x – 4)" 2nn²

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Q: 1.) Use the Limit Comparison Test to determine if the series converges or diverges. 1 sin().

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

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Q: What is the interval of convergence for power series 2" E (x +5)"? 3n+1 n=1

A: Given apply ratio test

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A: Given that f(x)=2-x The taylor formula is,…

Q: 4. Represent the function f(x) = as a power series centered at zero. %3D 5+x?

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Q: use the biomal series (x + 1)k = 1 + kx + or || < 1 to approximate [¹ k(k − 1) 2! -x² + k(k-1)(k −…

A: Given : x+1k =1 +kx+kk-12!x2+kk-1k-23!x3+......    x&lt;1 To find  :∫01x42+1dx

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A: Solution is given as below:

Q: n(x + 10)" (2n)! n=0

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Q: 3. The numerical series log O converges to a ptive number O diverges to+0. Odiverges to-0. O…

A: We can find whether the given numerical series converges or diverges using limit comparison test.

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A: Solve the following

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Q: Find the first five non-zero terms of power series representation centered at x=0 for the function…

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Q: 1 Given that x" with convergence in (-1, 1), find the power series for with center 1+ 9x7 n=0 0.

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Q: Compute the value of the improper integral. (If the integral diverges to oo, type oo; if the…

A: The improper integral is given I=∫1∞lnx17xdx. Let lnx=t then 1xdx=dt Also, the limit changes from…

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Q: Find the interval and radius of convergence for the power series (3x + 5)k 2k k=1

A: ∑k=1∞ 3x+5k2k

Q: 1 Given that x" with convergence in (-1, 1), find the power series 1- x n=0 1 with center 6. for -

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Q: What is the largest value of x for which the series (x- 3)" n°87 is absolutely convergent? Yanıt:

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Q: 2. Expand the 27 period f(x) function to the Fouriér Series, where: f(x) = sinx in the range (0< x…

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Q: 2. Find a binomial series for the function and determine its radius of conv S() = V1-x с.

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Q: Does the series from 0 to infinity of (-1)^k/(2k+1) converge or diverge?

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Q: 5. Expand the function f(x)=1- cosx in the power series by the powers of (x+ 7). Find fll3)(-n).

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Q: The interval of convergence of the power (x – 7) 3 series : is - n=0 4 < x < 10

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Q: Subtract the infinite series of ln(1 – x) from In(1 + x) to get a power series for: + x Σ In %3D n=1…

A: Given:

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Q: ト→2 approachest) limirof khe fallowing series as x is Feo = x-+瓷- ,3 X-X %D

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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: Q2) By using power series method only prove that:- cs dx = In( 1 + x) + CS; 1+ x

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Q: Compute accurate to 3 decimal places using binomial series the value of the definite integral. dx а.…

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Q: 4. If > Cn (T + 2)" converges for -5 < x < 1, find the radius of convergence of the series. n=1

A: The given power series is cn(x+2)n , n from 1 to infinity.

Q: Q2) By using power series method only prove that:- cs dx = In(1 + x) + CS 1+ x

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