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1. $$y = 2 − 3x − 2x^{2} + \frac{7}{2}x^{3} − \frac{55}{12}x^{4} + \frac{59}{8}x^{5} − \frac{83}{6}x^{6} + \frac{9547}{ 336}x^{7} + \ldots$$

2. $$y = −1 + 2x − 4x^{3} + 4x^{4} + 4x^{5} − 12x^{6} + 4x^{7} + \ldots$$

3. $$y = 1 + x^{2} − \frac{2}{3}x^{3} + \frac{11}{6}x^{4} − \frac{9}{5} x^{5} + \frac{329}{90}x^{6} − \frac{1301}{315} x^{7} + \dots$$

4. $$y = x − x^{2} − \frac{7}{2} x^{3} + \frac{15}{2} x^{4} + \frac{45}{8} x^{5} − \frac{261}{8}x^{6} + \frac{207}{16} x^{7} + \ldots$$

5. $$y = 4 + 3x − \frac{15}{4} x^{2} + \frac{1}{4} x^{3} + \frac{11}{16} x^{4} − \frac{5}{16} x^{5} + \frac{1}{20} x^{6} + \frac{1}{120} x^{7} + \ldots$$

6. $$y = 7 + 3x − \frac{16}{3} x^{2} + \frac{13}{3} x^{3} − \frac{23}{9} x^{4} + \frac{10}{9} x^{5} − \frac{7}{27} x^{6} − \frac{1}{9} x{7} + \ldots$$

7. $$y = 2 + 5x − \frac{7}{4} x^{2} − \frac{3}{16} x^{3} + \frac{37}{192} x^{4} − \frac{7}{192} x^{5} − \frac{1}{1920} x^{6} + \frac{19}{11520} x^{7} + \ldots$$

8. $$y = 1 − (x − 1) + \frac{4}{3} (x − 1)^{3} − \frac{4}{3} (x − 1)^{4} − \frac{4}{5} (x − 1)^{5} + \frac{136}{45} (x − 1)^{6} − \frac{104}{63} (x − 1)^{7} + \ldots$$

9. $$y = 1 − (x + 1) + 4(x + 1)^{2} − \frac{13}{3} (x + 1)^{3} + \frac{77}{6} (x + 1)^{4} − \frac{278}{15} (x + 1)^{5} + \frac{1942}{45} (x + 1)^{6} − \frac{23332}{315} (x + 1)^{7} + \ldots$$

10. $$y = 2 − (x − 1) − \frac{1}{2} (x − 1)^{2} + \frac{5}{3} (x − 1)^{3} − \frac{19}{12} (x − 1)^{4} + \frac{7}{30} (x − 1)^{5} + \frac{59}{45} (x − 1)^{6} − \frac{1091}{630} (x − 1)^{7} + \ldots$$

11. $$y = −2 + 3(x + 1) − \frac{1}{2}(x + 1)^{2} − \frac{2}{3}(x + 1)^{3} + \frac{5}{8}(x + 1)^{4} − \frac{11}{30} (x + 1)^{5} + \frac{29}{144} (x + 1)^{6} − \frac{101}{840 }(x + 1)^{7} + \ldots$$

12. $$y = 1 − 2(x − 1) − 3(x − 1)^{2} + 8(x − 1)^{3} − 4(x − 1)^{4} − \frac{42}{5}(x − 1)^{5} + 19(x − 1)^{6} − \frac{604}{35} (x − 1)^{7} + \ldots$$

19. $$y = 2 − 7x − 4x^{2} − \frac{17}{6}x^{3} − \frac{3}{4}x^{4} − \frac{9}{40}x^{5} + \ldots$$

20. $$y = 1 − 2(x − 1) + \frac{1}{2} (x − 1)^{2} − \frac{1}{6}(x − 1)^{3} + \frac{5}{36}(x − 1)^{4} − \frac{73}{1080}(x − 1)^{5} + \ldots$$

21. $$y = 2 − (x + 2) −\frac{7}{2}(x + 2)^{2} +\frac{4}{3}(x + 2)^{3} −\frac{1}{24}(x + 2)^{4} +\frac{1}{60}(x + 2)^{5} + \ldots$$

22. $$y = 2 − 2(x + 3) − (x + 3)^{2} + (x + 3)^{3} −\frac{11}{12}(x + 3)^{4} +\frac{67}{60}(x + 3)^{5} +\ldots$$

23. $$y = −1 + 2x + \frac{1}{3}x^{3} −\frac{5}{12}x^{4} + \frac{2}{5}x^{5} + \ldots$$

24. $$y = 2 − 3(x + 1) + \frac{7}{2} (x + 1)^{2} − 5(x + 1)^{3} + \frac{197}{24}(x + 1)^{4} − \frac{287}{20}(x + 1)^{5} + \ldots$$

25. $$y = −2 + 3(x + 2) − \frac{9}{2}(x + 2)^{2} +\frac{11}{6}(x + 2)^{3} +\frac{5}{24}(x + 2)^{4} +\frac{7}{20}(x + 2)^{5} + \ldots$$

26. $$y = 2 − 4(x − 2) − \frac{1}{2}(x − 2)^{2} + \frac{2}{9}(x − 2)^{3} + \frac{49}{432}(x − 2)^{4} + \frac{23}{1080} (x − 2)^{5} + \ldots$$

27. $$y = 1 + 2(x + 4) − \frac{1}{6}(x + 4)^{2} −\frac{10}{27}(x + 4)^{3} +\frac{19}{648}(x + 4)^{4} +\frac{13}{324}(x + 4)^{5} + \ldots$$

28. $$y = −1 + 2(x + 1) −\frac{1}{4}(x + 1)^{2} +\frac{1}{2}(x + 1)^{3} −\frac{65}{96}(x + 1)^{4} + \frac{67}{80}(x + 1)^{5} + \ldots$$

31.

1. $$y=\frac{c_{1}}{1+x}+\frac{c_{2}}{1+2x}$$
2. $$y=\frac{c_{1}}{1-2x}+\frac{c_{2}}{1-3x}$$
3. $$y=\frac{c_{1}}{1-2x}+\frac{c_{2}x}{(1-2x)^{2}}$$
4. $$y=\frac{c_{1}}{2+x}+\frac{c_{2}x}{(2+x)^{2}}$$
5. $$y=\frac{c_{1}}{2+x}+\frac{c_{2}}{2+3x}$$

32. $$y = 1 − 2x −\frac{3}{2} x^{2} + \frac{5}{3} x^{3} + \frac{17}{24} x^{4} − \frac{11}{20} x^{5} + \ldots$$

33. $$y = 1 − 2x − \frac{5}{2} x^{2} + \frac{2}{3} x^{3} − \frac{3}{8} x^{4} + \frac{1}{3} x^{5} + \ldots$$

34. $$y = 6 − 2x + 9x^{2} + \frac{2}{3} x^{3} − \frac{23}{4} x^{4} − \frac{3}{10} x^{5} + \ldots$$

35. $$y = 2 − 5x + 2x^{2} − \frac{10}{3} x^{3} + \frac{3}{2} x^{4} − \frac{25}{12} x^{5} + \ldots$$

36. $$y = 3 + 6x − 3x^{2} + x^{3} − 2x^{4} − \frac{17}{20} x^{5} + \ldots$$

37. $$y = 3 − 2x − 3x^{2} + \frac{3}{2} x^{3} + \frac{3}{2} x^{4} − \frac{49}{80} x^{5} + \ldots$$

38. $$y = −2 + 3x + \frac{4}{3} x^{2} − x^{3} − \frac{19}{54} x^{4} + \frac{13}{60} x^{5} + \ldots$$

39. $$y_{1}=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m}}{m!}=e^{-x^{2}},\:y_{2}=\sum_{m=0}^{\infty}\frac{(-1)^{m}x^{2m+1}}{m!}=xe^{-x^{2}}$$

40. $$y = −2 + 3x + x^{2} − \frac{1}{6} x^{3} −\frac{3}{4}x^{4} + \frac{31}{120}x^{5} + \ldots$$

41. $$y = 2 + 3x − \frac{7}{2} x^{2} − \frac{5}{6} x^{3} + \frac{41}{24} x^{4} + \frac{41}{120} x^{5} + \ldots$$

42. $$y = −3 + 5x − 5x^{2} + \frac{23}{6}x^{3} − \frac{23}{12}x^{4} + \frac{11}{30} x^{5} + \ldots$$

43. $$y = −2 + 3(x − 1) + \frac{3}{2}(x − 1)^{2} −\frac{17}{12}(x − 1)^{3} −\frac{1}{12}(x − 1)^{4} +\frac{1}{8}(x − 1)^{5} + \ldots$$

44. $$y = 2 − 3(x + 2) + \frac{1}{2}(x + 2)^{2} − \frac{1}{3}(x + 2)^{3} + \frac{31}{24}(x + 2)^{4} −\frac{53}{120}(x + 2)^{5} + \ldots$$

45. $$y = 1 − 2x + \frac{3}{2} x^{2} − \frac{11}{6} x^{3} + \frac{15}{8} x^{4} − \frac{71}{60} x^{5} + \ldots$$

46. $$y = 2 − (x + 2) − \frac{7}{2}(x + 2)^{2} − \frac{43}{6}(x + 2)^{3} − \frac{203}{24} (x + 2)^{4} − \frac{167}{30} (x + 2)^{5} + \ldots$$

47. $$y = 2 − x − x^{2} + \frac{7}{6} x^{3} − x^{4} + \frac{89}{120} x^{5} + \ldots$$

48. $$y = 1 + \frac{3}{2} (x − 1)^{2} + \frac{1}{6} (x − 1)^{3} − \frac{1}{8} (x − 1)^{5} + \ldots$$

49. $$y = 1 − 2(x − 3) + \frac{1}{2}(x − 3)^{2} − \frac{1}{6}(x − 3)^{3} + \frac{1}{4}(x − 3)^{4} − \frac{1}{6}(x − 3)^{5} + \ldots$$