Use the power series11 + x∞Σ (-1)^x^, |x| < 1n = 0to find a power series for the function, centered at 0.In(x6 + 1)f(x)∞==f(x) = |(−1)¹x6nn = 0XDetermine the interval of convergence. (Enter your answer using interval notation.)(−1,1)X

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Answered: Use the power series ∞ 1 1 + x =…

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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Use the power series 00 E(-1)^x" 1 + x n = 0 to determine a power series, centered at 0, for the function. 1 dx х+ 1 f(x) = In(x + 1) = f(x) = > n = 0 Identify the interval of convergence. (Enter your answer using interval notation.)
For the function ln(5x+11), (a) write a power series representation of the function, (b) find the center and radius of convergence of the power series from (a), and (c) find the interval of convergence of the same power series.
(2n+ 1)x Please explan Determine whether the series sin- converges or diverg es. If it converges, find its sum. thish L and how it nelotes to the n-D0 (4r + 1)x Since sin =1 and sin =-1 and the angle may be incremente by any multiple of 2x, then sin 2 =1 and (4r+3)a sin =-1, (any r). How does this nelate to the answer. Examine the partial sums of the series. The expanded series is sin 3n 5x + sin + sin ,+ sin + sin+ .. ansver Notice that the angles in consecutive even terms (n is even) differ by 2x, and the angles in consecutive odd terms (n is odd) differ by 2x. Since sin 0 = sin (0+2mx), all the even terms are equal to 1, and all the odd terms are equal to -1. Evaluating each term, the series is 1+(-1)+1+(-1)+1+ The partial sums altermate between 1 and 0, tending to (2n + 1)a neither value. Therefore, lim sin- k00 n=0 does not exist and the series diverges. detail. Pelease explain n thanke gou.

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Use the power series ∞ 1 1 + x = (-1)^x², |×| < 1 x n = 0 to find a power series for the function, centered at 0. f(x) = n(x6 + 1) (-1) ¹x6n 00 f(x) = Σ n = 0 Determine the interval of convergence. (Enter your answer using interval notation.) (−1,1)