Carry out the substitution x = sin 0 in the Taylor series 1.3.5..(2n - 1) x2n+1 arcsin x = x + 2 2.4.6..(2n) 2n +1 n=1 valid for |x| < 1 and use the formula • 7/2 2.4. (2n) 2n+ 0 de = ... 3.5.. (2n + 1)'

Q: Evaluate fr.co x cos(x³) dx as an infinite series.

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Q: +x (-1)" Given cos(x) = (-1)". n=0 (2n)! -x2n for all x E R, %3D 1. Express the integral cos x dx as…

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Q: . (a) Find the Taylor series of cos 2x at x = 0, either by direct calculation or by using the known…

A: a) The Taylor series expansion for cosx at x=0 is, cosx=1-12!x2+14!x4-16!x6+.... Replace x by 2x in…

Q: b) Evaluate S(2y + 4z)dx + (2æ + 3z)dy + (4x + 3y)dz, where C is the curve given by r(t) = for 0 <t…

A: given ∫C2y+4zdx+2x+3zdy+4x+3ydz where C is the curve given by rt=t3,2sinπt2,3cosπt2 for 0≤t≤1…

Q: 5. Find the Fourier series of |x| of period 27 and evaluate the sum 1 1+ 32 1 +... 52

A: 5)

Q: 4- X

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Q: Question 0.4. (5 Points) Compute 1 (-1)" (Hint: Is this related to the Taylor series of a particular…

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Q: 3. Find Taylor series for tan-1x about x =1 .

A: We find the Taylor series for tan-1x about x=1.

Q: Q4. Find the Fourier series for f(x)= A – x _in the interval (0, 27). Deduce that 111

A: Let fx=π-x To find the Fourier series for fx in the interval 0,2π and further deduce…

Q: 2. Find the sum of series 20 De-3r)(r + 1). r=5

A: Since, you have asked multiple question, we will solved the first question for you. If you want any…

Q: Find the taylor series for f(x) = -4ex about x=5

A: The Taylor series of a function f(x) is given by  f(x)=∑n=0∞fn(a)n!(x-a)n Here, the given function…

Q: 7a) Find the first four derivatives of f(x) = and evaluate at x = 1. 7b) Find a formula for the nth…

A: We are given : f(x)=1x2⇒f'(x)=-2x3⇒f''(x)=(-2)(-3)x4=6x4⇒f'''(x)=-24x5⇒f''''(x)=120x6Now, at…

Q: Q5: Given that ao = 2, a₁ = 1, and b₁ = 1 are defined on the interval [-, π], check the linear…

A: Let the function f(x) be defined in [−π , π] then the Fourier series of f(x) is of the…

Q: Q1) a. Find the Fourier series of 0. x², f(x) = 6 b. Use the above result to show that л² -<x<0…

A: Given function is The Fourier series of f(x) on  is given by                      ------------------…

Q: 9 Write the Taylor Series for f(x) = sin(x) at x = 12/12/2 as & Cn (x-7) n Find the first 5…

A: Taylor serues

Q: Find the ninth term of the Taylor series of the function f(x) = cos(2x at x = 4 8 1 А. 8! 8 B. – 256…

A: The Taylor's series expansion of a function f(x) about x=a is given by;

Q: DO 11/2 x cos x² clx

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Q: Find the Fourier cosine series

A: Fourier cosine series

Q: 1) Find the Taylor Series for sin(x)

A: Since you have posted multiple questions, according to our guidelines we have done first question…

Q: Write the taylor series cetered at c = 1 for f(x) = 3x

A: The Taylor series of function f(x) centered at x=a is given by: f(x)≈∑n=0∞f(n)(a)n!(x-a)n

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Q: 1. Find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a. a) f(x) = e²x, a = 0 b)…

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

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

Q: (14). . Find the first fuer terms of the Taylor Series for the function flx)= ex9 ○メ=o.

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Q: Prove that 2e 5 2 sin (Tx/4)+1/2 dx ≤ 2e-1 1+2√2 (i) Show that zloga 7/4 dx - (log) (log x)-3/4 (ii)…

A: All the parts are solved properly with detailed explanation.Hope you got your answer. If any query…

Q: Q.1 Find the Taylor series of the cubic function x3 about x=3. Write the final formula of the…

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