Suppose the power series
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- (-1)"+1( n+1 (x-5)" 2. Consider the power series n=1 4n n What is the center ? What is the interval of convergencearrow_forward4. For the function f(x) : find the power series representation and its interval of (1 – 4.x)? - convergence.arrow_forwardIf xo is in the interval of convergence of a power series n=1 an (x – c)" that represents a function f(x), which of the following is true? Circle all that apply. (a) The value of f(xo) is approximated by the power series evaluated at x0, and the remain- der is non-zero. (b) The value of f(xo) is exactly equal to the power series evaluated at xo. (c) The interval of convergence must be all real numbers. (d) The only point of convergence is the center of the power series x = c. If f(x) is represented by a power series E-, an (x – c)" over an open interval containing c, then which of the following is true about that power series? Circle all that apply. (a) The remainder of the power series is less than or equal to an. (b) The derivative of the power series is a power series. (c) The interval of convergence must be all real numbers. (d) It must be a Taylor series. (e) The radius of convergence is half the length of the interval of convergence.arrow_forward
- 3. Suppose that the power series c(-2)" converges for z=6 and diverges for r=-6. What must be true about the radius of convergence R n=0arrow_forwardFind the first few coefficients (c0, c1, c2, c3, c4,) and radius of convergence (R).arrow_forward6. Consider a power series of the form Σen (x-a)n n=0 with radius of convergence R. If a 0 and a #r, which of the following are possible intervals of convergence? (a) [-R, R) (b) (R-a, R+ a) (c) (a R/2, a + R/2) (d) (a R, a + R] (e) None of the above.arrow_forward
- 2. (25.10) xn (a) Show Σ 1n converges for x € [0, 1). E 1+xn (b) Show that the series converges uniformly on [0, a] for each a with 0 < a < 1. (c) Does the series converge uniformly on [0, 1)? Explain.arrow_forwardDetermine the interval of convergence for the power series representation of the following function. f(x)= 4x-3 6-x ○ (-4, 4) ○ (-1. 1) 아히 (-6,6) 아울휘 < Previous 대 3 No new data to save. Last checked at 8:3arrow_forward2. I) Find the centre, radius, and interval of convergence for the given power series. Σ Е (13)" nlnn 1 -(x+96)³" 3n n=2 (II) Determine all values of x where the given power series (a) absolutely convergent and (b) conditionally convergent.arrow_forward
- le) (b) 2E" 2" 12. Check the convergence of the following power series, find the radius and interval of conver- gence: ( a) Σ" (x+ n)" ( b) Σ Also find the sum as a function of x for this problem. 13. Find the Taylor series generated by ƒ at r = a and the Maclaurin series: (a) f(x) = 1/x², a = 1 (b) f(x) = xª + x² + 1, a = -2 14. Find the Taylor polynomials of order 1,2,3 for the function f at a: (a) f(x)= /T, a = 4 (b) f(r) = cos r, a = r /4 15. (a) Calculate e with an error of less than 10-º. (b) Estimate the error in the approximation sinh(x) = x + (r"/3!) when |r| < 0.5. (c) How close is the approximation sin(x) = r when |æ| < 10¬³? For which of these values of r is a< sin(x)? %3Darrow_forwardJj.189.arrow_forward3. Show that the power series (a)-(c) have the same radius of convergence. Then show that (a) diverges at both endpoints, (b) converges at one endpoint but diverges at the other, and (c) converges at both endpoints. x" п3 x" (c) n23n (a) x" (b) n=1 n=1 n=1arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
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