Series@Sin@xD, 8x, 0, 4<D

x -

x3

6

+ O@xD5

Series@Sin@xD, 8x, 1, 6<D

Sin@1D + Cos@1D Hx - 1L -

1

2

Sin@1D Hx - 1L2-

1

6

Cos@1D Hx - 1L3+

1

24

Sin@1D Hx - 1L4+

1

120

Cos@1D Hx - 1L5-

1

720

Sin@1D Hx - 1L6+ O@x - 1D7

Normal@%D

H-1 + xL Cos@1D -

1

6

H-1 + xL3Cos@1D +

1

120

H-1 + xL5Cos@1D +

Sin@1D -

1

2

H-1 + xL2Sin@1D +

1

24

H-1 + xL4Sin@1D -

1

720

H-1 + xL6Sin@1D

Plot@%, 8x, 0, 10<D

2 4 6 8 10

-80

-60

-40

-20