Q1. a) Evaluate the complex integral, around a closed contour C, where vertices of C are +2, +2i [CLO2] e-z -dz -4) (2-필) b) Is the following series convergent or divergent? Give a reason. [CLO2] :(1+ i)" (Зп)! n=0

Q: R, is the error term in approximating f(x) by taking the sum of the first (n + 1) terms of its…

A: Given that, fx=In1+x The Maclaurin series for given function is,…

Q: Using the nth term test for divergence, the series 2n n +5 an n31 n-1 is divergent since lim an =

A: According to nth term test for divergent, The given series is,

Q: Evaluate the indefinite integral as a power series. C+ n-0 What is the radius of convergence R? R =

A: Remainder: 11-X=∑n=0∞Xn So,  t1-t9=t11-t9=t∑n=0∞t9n=∑n=0∞t9n+1

Q: Evaluate the indefinite integral as a power series. x² In(1 + x) dx f(x) = C + E What is the radius…

A: To evaluate as a power series: ∫x2ln1+xdx We can write ln1+x as ∫11+xdx ⇒ln1+x = ∫11--xdx Writing…

Q: The following sum 2n 4 - +...+ 4- is a right Riemann sum with n subintervals of equal length for the…

A: Given definite integral∫abf(x)dxWe know that the right Riemann sum with the n subintervals of equal…

Q: Use the to Series integral test whe ther the de termine -15 convergent or divergent n.Ln(n) n=2

A: The given series is ∑n=2∞5n.lnn

Q: Evaluate the indefinite integral as a power series. tan(x) dx 00 f(x) = C + %3D n = 0 What is the…

A: From the power series we can say that:

Q: Given a) Prove the three conditions required of the corresponding function f(x) that must be…

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Q: Using the nth term test for divergence, the series an V52n+1 n%3D1 is divergent since lim a,

A: Explanation of the answer is as follows

Q: Use the Integral Test to determine whether the series is convergent or divergent. Vn + 6 n = 1…

A: see 2nd step

Q: Use the Integral Test to determine whether the series is convergent or divergent. n2 + 3 n = 1…

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Q: Evaluate the indefinite integral as a power series t dt 1 - t C+ £ n = 0 What is the radius of…

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Q: 12. Consider the function f(x) = -x³ + 3x2 + 12x – 2 on the interval [-4,6]. %3D Using left Riemann…

A: In this question we will use basic of calculus i.e use of right riemann sum and basic of definite…

Q: Consider the series n In(5n) n=2 a) Enter the integral whose convergence or divergence is equivalent…

A: Let f be a positive, decreasing function. Then the series ∑n=1∞f(n)  converges iff the the…

Q: (b) Without using the series form of Bessel's functions, show that a (x* J.(x)] = x° Je-1(x)

A: ANSWER 3 (b): NOTE: Just replace v by c We are asked to prove the result on derivative of Bessel…

Q: Evaluate the indefinite integral as an infinite series. dx n = 1

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Q: Write down the Taylor's series expansion of f(x) = cos x at x = in %3D terms of f(x), and its…

A: Hence,

Q: sin( 4. Classify the singularity of f(z) = at z = 0 as one of {removable singularity, essential…

A: Given function is  fz=sinz3-z3z13 The given function is analytic everywhere except at z=0 Therefore,…

Q: Evaluate the indefinite integral as a power series. t dt 1 - 7 C + n = 0 What is the radius of…

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Q: Evaluate the indefinite integral as a power series. x6 In(1 + x) dx f(x) = C + > x „6 (-1)"-x"+1 n+1…

A: The logarithm series formula states that ln1+x=∑n=0∞-1nxn+1n+1, x<1. Substitute ∑n=0∞-1nxn+1n+1…

Q: Use the integral test to determine the convergence or divergence of the series. 00 cos 1/k 21) E k2…

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Q: Evaluate the indefinite integral as a power series. dt 1- t9 C+ £ n = 0 What is the radius of…

A: The given problem is to find evaluate the given indefinite integral and express in power series and…

Q: a) Use integral test to determine whether the series d dx (tan ¹x) = Hint : 1 x² + 1 k=1 tan k e k²…

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Q: (a) |r cos(a*)dz

A: As per our guidelines we are supposed to answer only first part. Kindly repost other parts in the…

Q: Evaluate the indefinite integral as a power series. x4 In(1 x) dx Σ f(x) C n 1 What is the radius of…

A: Consider the given indefinite integral.

Q: too for all x in R. n! n=0 class, you saw that e" Represent the function f(r) = xe as a power series…

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Q: 5. Suppose that f(x) = sin x on the interval [0,2n]. Partition the interval [0, 2n] into four…

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Q: (z + j)" n2 n=1

A: series and sequences

Q: Use the Integral Test to determine whether the series is convergent or divergent. Σ n2 + 6 n = 1…

A: Given data: The expression for the given series is ∑n=1∞nn2+6.   The expression for the integral…

Q: only need to consider those series that are centered at the origin.) 1 f(z) = z²(1+z)² (b) What is…

A: We have to find the residues of the given function.

Q: 13) Evaluate the integral as an infinite series dt

A: ∫e3t2 dtSubstitute t=3t → dudt=3 →dt=13duDivide and multiply the integral by π

Q: 1 2n 88. f (x) = %3D %3D 1-x2 n=0

A: Differenciate the given series with respect to x

Q: Use the Integral Test to determine whether the infinite series is convergent. 00 Σ 1 n² + 25 n=5…

A: To use integral test to determine the infinite series is convergent  ∑n=5∞1n2+25

Q: 2. Determine the Maclaurin Series (5th approximation) of the function defined by f(x) = sin x .

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Q: Use the Integral Test to determine whether the series is convergent or divergent. n°e-n5 n = 1…

A: Given, ∑n=1∞n4e-n5

Q: Express the function f(r) as a power series centered at r = 0.

A: Note: "Since you have asked multiple questions, we will solve the first question for you. If you…

Q: ind the Taylor series of f(x)=sin(x) centered at a=π.

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Q: Use a similar p-Integral to determine whether the infinite series is convergent. n Fill in the…

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Q: Consider the series Vk 6 + 6k k=1 a) Enter an integral whose convergence or divergence is equivalent…

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Q: Calculate the limited expansion centered on the origin, of order n for f(x) = cosh(x) + sinh(x²)…

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Q: Use the Integral Test to determine whether the infinite series is convergent. 12ne T-1 Fill in the…

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Q: PExpand the function f(x) = {," s-1, -n < x < 0 0 < x <n in a Fourier series. To what value does the…

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Q: Use the Integral Test to determine whether the series is convergent or divergent. n = 1 (2n + 5)3…

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Q: Quy assume f(x) has the Taylor series write your aswer without rouding 4-2" (x-4)" (24)! h=0 or…

A: Given :  f(x) has the Taylor series  ∑n=0∞4.2n(2n)! x-4n. To find :  f'''(4) without rounding or…

Q: Evaluate the indefinite integral as a power series. [x² x8 In(1 + x) dx 8 f(x) = c + Σ n = 1 What is…

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Q: Using the nth term test for divergence, the series Σ 32n+1 an = n=1 n=1 is divergent since lim an n→…

A: We need to evaluate given limit

Q: Use the Integral Test to determine whether the infinite series is convergent. A n+4 n=2 Fill in the…

A: We have to find out the convergence test

Q: Evaluate the indefinite integral as a power series. dt 1 t5 C + n = 0 What is the radius of…

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Q: Use series to compute the indefinite integral. | 2x cos(x4) dx Σ! n = 0

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Q: Using the nth term test for divergence, the series Σ 32n+1 n=1 n=1 is divergent since lim an

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