2e- is 1 /4 – x² Show that the Maclaurin series for .. 4 Hence, 2e- 1 V4 – x² i. evaluate lim 2e- ii. approximate the definite integral o1 z4- 72 0.4 dx correct to 4 decimal x/4 – x² places.

Q: cos x dx - In (1 +x) 1- cos x a) b) lim

A: Given: a)∫011-cos xx2dx b)limx→0ln1+x21-cos x

Q: 35a. Use long division to derive the first four non-zero terms of the Maclaurin Series for f(x) =…

A: The required Maclaurin Series for the function f(x)=tan x are: 1.  sin x=x-x33!+x55!-x77!+x99!-...…

Q: sin z z7

A:

Q: V. Find the exponential Fourier transform g(a) of the given function f(x) and write it as a Fourier…

A: Using the given integral formula:

Q: Q3: A. By power series, prove that: d Cosx = -sinx dx B. Find the value of x for which Inx + (Inx)²…

A:

Q: Q. 29. Calculate harmonic from the following continuous series: Class Interval : 0-10 10-20 20-30…

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Q: /3 In (x + 1) The series representing the integral - dx is given by OA. (-1)"+1 3" n2 OB. (-1)"+1…

A: To find: Series representation of ∫013lnx+1xdx

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Q: Q- het m = 10-81 +3. compute the Suron E Sin (mntt) cos (m +2) (m+1)"+ COS n+1 n=1

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Q: IV. Approximate the following up to 4 decimal places using Maclaurin's series... 1. sin (110) 2.…

A:

Q: Evaluate this indefinite integral as a power series.

A: According to the given information, it is required to determine the power series of indefinite…

Q: Q.2\ The series E is n(1+ln(n))

A: ∑n=1∞1n1+lnn

Q: 6. Find the sine series of Sx, f(x) 0 <x < #/2 - x, T/2 < x < T

A:

Q: (-1)"x2" (2n)! +00 1. For all x ER, cos x = Σ n=0 a. Find a power series that is equal to x cos(x)…

A: Given that, cos x = ∑n = 0∞(-1)nx2n2n! a) Find the power series equal to x cos x2 b) Differentiating…

Q: 6.3 Represent the function f(x) =,, using a power series and find the interval of convergence. 2 – x

A:

Q: r1/10 Approximate the definite integral dz to within an error of tat most 10-6. Hint: Use the…

A: Let I = ∫0110 e-x2 dx

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Q: For each n ≥ 2, let an S = an+1 an anan+1 4n³-6n 9n² - 5n - 6 1 -sin n(). Evaluate the following…

A: The nth term of a sequence {an}n≥2 is given by  an=4n3-6n9n2-5n-6sinπn We have to evaluate…

Q: III. In each of the following, express the required series in summation notation. 1. Obtain the…

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Q: 1 x" (а) п1 — х1 n=1 (-1)" (b) E x2 – n² n=1 8.

A:

Q: 3. If we know that > ak = 10,000, then what can we way about lim ay? %3D k=1

A: Explanation of the answer is as follows

Q: Fin:d the Fot:ier sine seri:s in P T/3, for 0<x<T/3, f(x)= 0, for T/3<x<2r/3, -7/3, for 2n/3<x<r.…

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Q: 2b. Find the range of value of x for which the following series is convergent or divergent 2'x? 22x3…

A: The given series is x1+21x22+22x33+23x44+... Here an=2n-1xnn We will check its convergence by…

Q: Use the standard series to find the sixth order approximation to f(x, y) = centered 1- xy at (0,0).…

A: The given function is similar to the infinite geometric series, which is as follows:…

Q: Q5. Let the improper integrals, sin()dx and sin² ()dx (i) (ii) Write down the first 3 terms of…

A: Introduction: Sometimes the Maclaurin series is very efficient to integrate a function. The power…

Q: 7. Use the Maclaurin series for cos x to find lim x-0 1- cos x X

A: Given: limx→01-cosxxWe need to find the given limit using the Maclaurin series.

Q: 4. Show that tf ftz) = sinz, then fom0) 0 and f0) (-1)" (a0.1.2...). Thus give an alternative…

A: In the given problem,   we have to show that if fz = sin z , then f2n0 = 0 and f2n+10 = -1n , n = 0…

Q: Evaluate the indefinite integral as an infinite series. | arctan(x2) dx Σ + C n = 0

A: Answer