1. Find the radius of convergence of:k1+2i(a) Σ(-1)* (¹ + ²i) * (z +2i)*.2k=0(b)(-1) kk2k (z − 1 - i)k.k=1(©) Σ (³ + (−1)*) * (²) *.k=0

1) (a) The given series is Suppose that Then, Then, the radius of convergence is (b) The…

Answered: 1. Find the radius of convergence of: k…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Q(A). Let {fn(x)}- be a sequence of functions In=1 1+ (x – 2)“ J n=1 defined over [2,3]. Show that: (a) fn(x) is meaurable and monotonic increasing for all n. (b) {fn(x)}=1 converges a.e. to a function f(x) to be determined. 3 (c) Apply the monotone convergence theorem to evaluate Lim | fn(x)dµ. n 00
hn (x) = n(x²+1) for x E R and nEN. For Example Prove that hn is uniformly convergent on R.
-4h (a) Derive the Gaussian quadrature formula of the form fh f(x) dx = a f(h) + bf (2h) + cf (3h). Find the order of convergence of this formula. (b) Apply the formula you found on part (a) to X₁+k = x₁ + fti+k f(t, x(t))dt, to find an approximate formula for the numerical solution of the IVP (t) = f(t, x(t)), x(t₁) = xo. (c) Use the IVP (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the interval [0, 2] with h = 0.25. [Hint: the exact solution is x(t) = (2) t²-t.
6. L Graph y sin z and the Taylor polynomial T₂M+1(x) = M (-1)"2n+1 (2n + 1)! 700 on the interval (-2, 2n) for M = 1, 2, 3, ..., until you find a value of M for which there's no perceptible difference between the two graphs.
(17) a) Find the third order Taylor polynomial T;(x) for f(x)= In x centered at a=1; identify the interval and radius of convergence. b) Use your answer in a) to find an approximation to In(1.05). c) Using the nth remainder term for Taylor's Theorem, R,(x)|< M (x-1)*1 , and the (n+1)! graph of the fourth derivative of Inx given below determine the accuracy of your answer in part b). 1 1.02 1.04 --2- -4. -6-
Which of the following statements is CORRECT? O Gauss-Jordan Elimination is more complex than Gaussian Elimination. Computers can compute the root for f(x) = x2 - 14 = 0 with an actual error equals to zero. O None O Our calculations for computing the area of a circle (with radius r) is always exact using the relation O The rate of convergence of Jacobi method is more rapid than Gauss-seidel method.
3.) Using an iterative such as Gauss -Seidel, solve the converged results at three decimal places. -3x + 4y + 5z = 6 -2x + 2y - 3z = -3 2y - z = 1
2
2. (20 points) The function defined by f(x) = cos(x) has zeros at every odd integer. Which zero off does the Bisection method converge when applied on the following each interval. (a) [-1.1, 3.3] (b) [a, b], where -1 < a < 1 and 5
(c) Explain the Bisection Method to compute a real root of f(x) = 0 and find the condition of convergence. Hence, find a non-zero of the equation x² + 4 cos x = 0.
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Which converge and diverge?

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1. Find the radius of convergence of: k (a) (−1)k (¹+2i) * (z +2i)*. k=0 k (b) Ỹ (−1)² (z − 1 − i)*. - k2k (e) Σ (3+(-1)*)* (²) *. k=0

1. Find the radius of convergence of: k (a) (−1)k (¹+2i) * (z +2i). k=0 k (b) Ỹ (−1)² (z − 1 − i). - k2k (e) Σ (3+(-1)) (²) *. k=0