Page 858 - Advanced_Engineering_Mathematics o'neil

838 Answers to Selected Problems

5.
∞
16 1
sin((2n − 1)x),
π 2n − 1
n=1
converging to −4for −π< x < 0, to 4 for 0 < x <π andto0for x = 0,−π,π.
7.
∞
13 16 nπx 4 nπx
+ (−1) n cos + sin ,
2
3 n π 2 2 nπ 2
n=1
converging to f (x) for −2 < x < 2, to 2 at x =−2and to7at x = 2.
9.
∞
3 1 1 − (−1) n
+ sin(nx),
2 π n
n=1
converging to 1 for −π< x < 0, to 2 for 0 < x <π and to 3/2at x = 0,−π,π.
11.
∞
1 (−1) n+1 nπx
sin(3) + 6sin(3) cos ,
3 n π − 9 3
2
2
n=1
converging to cos(x) on [−3,3].
13. The series converges to 3/2for x =±3, to 2x if −3 < x < −2, to −2if x =−2, to 0 if −2 < x < 1, to 1/2if x = 1and
2
to x if 1 < x < 3.
2
2
15. The series converges to (2 + π )/2if x =±π,to x if −π< x < 0,to1if x = 0and to2if0 < x <π.
17. The series converges to −1if −4 < x < 0, to0if x =±4or x = 0,andto1if0 < x < 4.
19. The series converges to
⎧
⎪−1 for x =−4and for x = 4
⎪
⎪
3/2 for x =−2
⎨
⎪5/2 for x = 2
⎪
⎪
f (x) elsewhere on [−4,4].
⎩
Section 13.3 Sine and Cosine Series
1. The cosine series is 4, the function itself, for 0 ≤ x ≤ 3. The sine series is
∞
16 1 (2n − 1)πx
sin ,
π 2n − 1 3
n=1
converging to 0 if x = 0or x = 3andto 4for0 < x < 3.
3. The cosine series is
∞ n
1 2 (−1) (2n − 1) (2n − 1)x
cos(x) − cos ,
2 π (2n − 3)(2n + 1) 2
n=1
converging to 0 for 0 ≤ x <π,to0at x = 2π,tocos(x) for π< x < 2π,and to −1/2at x = π.
The sine series is
∞
−2 −2n
n
sin(x/2) + (cos(nπ/2) + (−1) )sin(nx/2),
3π π(n − 4)
2
n=3
converging to 0 for 0 ≤ x <π and for x = 2π,to −1/2for x = π, and to cos(x) for π< x < 2π.
5. The cosine series is
∞
4 16 (−1) n
+ cos(nπx/2),
3 π 2 n 2
n=1
2
converging to x for 0 ≤ x ≤ 2. The sine series is
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October 14, 2010 17:50 THM/NEIL Page-838 27410_25_Ans_p801-866