1. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}(2m+1)x^{2m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}(m+1)x^{2m+1}\)
2. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m+1}\frac{x^{2m}}{2m-1}+a_{1}x\)
3. \(y=a_{0}(1-10x^{2}+5x^{4})+a_{1}\left(x-2x^{3}+\frac{1}{5}x^{5}\right)\)
4. \(y=a_{0}\sum_{m=0}^{\infty}(m+1)(2m+1)x^{2m}+\frac{a_{1}}{3}\sum_{m=0}^{\infty}(m+1)(2m+3)x^{2m+1}\)
5. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod _{j=0}^{m-1}\frac{4j+1}{2j+1} \right]x^{2m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}(4j+3) \right]\frac{x^{2m+1}}{2^{m}m!} \)
6. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}\frac{(4j+1)^{2}}{2j+1} \right]\frac{x^{2m}}{8^{m}m!}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}\frac{(4j+3)^{2}}{2j+3} \right]\frac{x^{2m+1}}{8^{m}m!}\)
7. \(y=a_{0}\sum_{m=0}^{\infty}\frac{2^{m}m!}{\prod_{j=0}^{m-1}(2j+1)}x^{2m}+a_{1}\sum_{m=0}^{\infty}\frac{\prod_{j=0}^{m-1}(2j+3)}{2^{m}m!}x^{2m+1}\)
8. \(y=a_{0}\left(1-14x^{2}+\frac{35}{3}x^{4} \right)+a_{1}\left(x-3x^{3}+\frac{3}{5}x^{5}+\frac{1}{35}x^{7} \right)\)
9. (a) \(y=a_{0}\sum_{m=0}^{\infty} (-1)^{m}\frac{x^{2m}}{\prod_{j=0}^{m-1}(2j+1)}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{2m+1}}{2^{m}m!}\)
10. (a) \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}\frac{4j+3}{2j+1} \right]\frac{x^{2m}}{2^{m}m!}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}\frac{4j+5}{2j+3} \right]\frac{x^{2m+1}}{2^{m}m!}\)
11. \(y=2-x-x^{2}+\frac{1}{3}x^{3}+\frac{5}{12}x^{4}-\frac{1}{6}x^{5}-\frac{17}{72}x^{6}+\frac{13}{126}x^{7}+\ldots \)
12. \(y=1-x+3x^{2}-\frac{5}{2}x^{3}+5x^{4}-\frac{21}{8}x^{5}+3x^{6}-\frac{11}{16}x^{7}+\ldots \)
13. \(y=2-x-2x^{2}+\frac{1}{3}x^{3}+3x^{4}-\frac{5}{6}x^{5}-\frac{49}{5}x^{6}+\frac{45}{14}x^{7}+\ldots \)
16. \(y=a_{0}\sum_{m=0}^{\infty}\frac{(x-3)^{2m}}{(2m)!}+a_{1}\sum_{m=0}^{\infty}\frac{(x-3)^{2m+1}}{(2m+1)!}\)
17. \(y=a_{0}\sum_{m=0}^{\infty}\frac{(x-3)^{2m}}{2^{m}m!}+a_{1}\sum_{m=0}^{\infty}\frac{(x-3)^{2m+1}}{\prod_{j=0}^{m-1}(2j+3)}\)
18. \(y=a_{0}\sum_{m=0}^{\infty}\left[\prod_{j=0}^{m-1}(2j+3) \right]\frac{(x-1)^{2m}}{m!}+a_{1}\sum_{m=0}^{\infty}\frac{4^{m}(m+1)!}{\prod_{j=0}^{m-1}(2j+3)}(x-1)^{2m+1}\)
19. \(y=a_{0}\left(1-6(x-2)^{2}+\frac{4}{3}(x-2)^{4}+\frac{8}{135}(x-2)^{6}\right)+a_{1}\left((x-2)-\frac{10}{9}(x-2)^{3}\right)\)
20. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}(2j+1) \right]\frac{3^{m}}{4^{m}m!}(x+1)^{2m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\frac{3^{m}m!}{\prod_{j=0}^{m-1}(2j+3)}(x+1)^{2m+1}\)
21. \(y=-1+2x+\frac{3}{8}x^{2}-\frac{1}{3}x^{3}-\frac{3}{128}x^{4}-\frac{1}{1024}x^{6}+\ldots \)
22. \(y = −2 + 3(x − 3) + 3(x − 3)^{2} − 2(x − 3)^{3} − \frac{5}{4}(x − 3)^{4} + \frac{3}{5}(x − 3)^{5} + \frac{7}{24}(x − 3)^{6} − \frac{4}{35}(x − 3)^{7} + \ldots \)
23. \(y = −1 + (x − 1) + 3(x − 1)^{2} − \frac{5}{2}(x − 1)^{3} − \frac{27}{4}(x − 1)^{4} + \frac{21}{4}(x − 1)^{5} + \frac{27}{2}(x − 1)^{6} − \frac{81}{8}(x − 1)^{7} + \ldots \)
24. \(y = 4 − 6(x − 3) − 2(x − 3)^{2} + (x − 3)^{3} + \frac{3}{2}(x − 3)^{4} − \frac{5}{4}(x − 3)^{5} − \frac{49}{20}(x − 3)^{6} + \frac{135}{56}(x − 3)^{7} + \dots\)
25. \(y = 3 − 4(x − 4) + 15(x − 4)^{2} − 4(x − 4)^{3} + \frac{15}{4}(x − 4)^{4} − \frac{1}{5}(x − 4)^{5}\)
26. \(y = 3 − 3(x + 1) − 30(x + 1)^{2} + \frac{20}{3}(x + 1)^{3} + 20(x + 1)^{4} − \frac{4}{3}(x + 1)^{5} − \frac{8}{9} (x + 1)^{6}\)
27.
- \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}x^{2m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}x^{2m+1}\)
- \(y=\frac{a_{0}+a_{1}x}{1+x^{2}}\)
33. \(y=a_{0}\sum_{m=0}^{\infty}\frac{x^{3m}}{3^{m}m!\prod_{j=0}^{m-1}(3j+2)}+a_{1}\sum_{m=0}^{\infty} \frac{x^{3m+1}}{3^{m}m!\prod_{j=0}^{m-1}(3j+4)}\)
34. \(y=a_{0}\sum_{m=0}^{\infty}\left(\frac{2}{3} \right)^{m}\left[\prod_{j=0}^{m-1}(3j+2) \right]\frac{x^{3m}}{m!}+a_{1}\sum_{m=0}^{\infty}\frac{6^{m}m!}{\prod_{j=0}^{m-1}(3j+4)}x^{3m+1}\)
35. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\frac{3^{m}m!}{\prod _{j=0}^{m-1}(3j+2)}x^{3m}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\left[\prod_{j=0}^{m-1}(3j+4) \right]\frac{x^{3m+1}}{3^{m}m!}\)
36. \(y=a_{0}(1-4x^{3}+4x^{6})+a_{1}\sum_{m=0}^{\infty}2^{m}\left[\prod_{j=0}^{m-1}\frac{3j-5}{3j+4} \right]x^{3m+1}\)
37. \(y = a_{0}\left( 1 + \frac{21}{2}x^{3} + \frac{42}{5}x^{6} + \frac{7}{20}x^{9}\right) + a_{1}\left(x + 4x^{4} + \frac{10}{7} x^{7}\right)\)
39. \(y=a_{0}\sum_{m=0}^{\infty}(-2)^{m}\left[\prod_{j=0}^{m-1}\frac{5j+1}{5j+4} \right]x^{5m}+a_{1}\sum_{m=0}^{\infty}\left(-\frac{2}{5}\right)^{m}\left[\prod_{j=0}^{m-1} (5j+2) \right]\frac{x^{5m+1}}{m!}\)
40. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{4m}}{4^{m}m!\prod_{j=0}^{m-1}(4j+3)}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{4m+1}}{4^{m}m!\prod_{j=0}^{m-1}(4j+5)}\)
41. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{7m}}{\prod_{j=0}^{\infty}(7j+6)}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{7m+1}}{7^{m}m!}\)
42. \(y=a_{0}\left(1-\frac{9}{7}x^{8}\right) +a_{1}\left(x-\frac{7}{9}x^{9}\right)\)
43. \(y=a_{0}\sum_{m=0}^{\infty}x^{6m}+a_{1}\sum_{m=0}^{\infty}x^{6m+1}\)
44. \(y=a_{0}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{6m}}{\prod_{j=0}^{m-1}(6j+5)}+a_{1}\sum_{m=0}^{\infty}(-1)^{m}\frac{x^{6m+1}}{6^{m}m!}\)
