(2n + 1)x Please explan thisn Determine whether the series 2 sin- converges or diverg as. If it converges, find its sum. n=0 (4r+1)x 31 and Land howit nelates to the Since sin 1 and sin =-1 and the angle may be incrementec by any multiple of 2x, then sin (4r+3)x sin =-1, (any r). tow does this nelate to the ansuer? Examine the partial sums of the series. The expanded series is sin + sin+ sin 5x + sin,+ sin +.... 2 ansue 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 e = sin (8+2mx), all the even terms are equal to 1, and al the odd terms are equal to -1. Evaluating each term, the series is 1+(-1)+1+(-1)+1+. The partial sums altemate between 1 and 0, tending to (2n+ 1)x neither value. Therefore, lim E sin- k00 n=0 does not exist and the series diverges.

(2n + 1)x Please explan thisn Determine whether the series 2 sin- converges or diverg as. If it converges, find its sum. n=0 (4r+1)x 31 and Land howit nelates to the Since sin 1 and sin =-1 and the angle may be incrementec by any multiple of 2x, then sin (4r+3)x sin =-1, (any r). tow does this nelate to the ansuer? Examine the partial sums of the series. The expanded series is sin + sin+ sin 5x + sin,+ sin +.... 2 ansue 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 e = sin (8+2mx), all the even terms are equal to 1, and al the odd terms are equal to -1. Evaluating each term, the series is 1+(-1)+1+(-1)+1+. The partial sums altemate between 1 and 0, tending to (2n+ 1)x neither value. Therefore, lim E sin- k00 n=0 does not exist and the series diverges.

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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