]> Exercise 2.12

Exercise 2.12

Set up a spreadsheet to compute the sin x for any inputted x . How many terms in the sin x power series expansion do you need to evaluate sin .5 to 8 decimal places?

Solution:

The power series expansion of sin x consists of the odd power terms in the expansion of exp x , with alternating signs,

sin x = x x 3 3 ! + x 5 5 !

We can set up a spreadsheet to compute it in many ways, and here is one.

We will do almost the same thing as done for exp x in column A we will put j ; and j will start at 1 and go up by 2 in each successive row. We will put x in the first row of column B and x 2 in successive rows, and multiply previous C entry by entry in B and divide by entry in A and by entry in A, minus 1.

We will get the following formulae in the spreadsheet

x

0.5

sin x  

1

=B1

=A2*B2

=D1+C2

=A2+2

=-B2*B2

=C2*B3/A3/(A3-1)

=D2+C3

=A3+2

=B3

=C3*B4/A4/(A4-1)

=D3+C4

=A4+2

=B4

=C4*B5/A5/(A5-1)

=D4+C5

=A5+2

=B5

=C5*B6/A6/(A6-1)

=D5+C6

=A6+2

=B6

=C6*B7/A7/(A7-1)

=D6+C7

=A7+2

=B7

=C7*B8/A8/(A8-1)

=D7+C8

=A8+2

=B8

=C8*B9/A9/(A9-1)

=D8+C9

=A9+2

=B9

=C9*B10/A10/(A10-1)

=D9+C10

=A10+2

=B10

=C10*B11/A11/(A11-1)

=D10+C11

=A11+2

=B11

=C11*B12/A12/(A12-1)

=D11+C12

=A12+2

=B12

=C12*B13/A13/(A13-1)

=D12+C13

=A13+2

=B13

=C13*B14/A14/(A14-1)

=D13+C14

=A14+2

=B14

=C14*B15/A15/(A15-1)

=D14+C15

=A15+2

=B15

=C15*B16/A16/(A16-1)

=D15+C16

=A16+2

=B16

=C16*B17/A17/(A17-1)

=D16+C17

=A17+2

=B17

=C17*B18/A18/(A18-1)

=D17+C18

=A18+2

=B18

=C18*B19/A19/(A19-1)

=D18+C19

=A19+2

=B19

=C19*B20/A20/(A20-1)

=D19+C20

=A20+2

=B20

=C20*B21/A21/(A21-1)

=D20+C21

=A21+2

=B21

=C21*B22/A22/(A22-1)

=D21+C22

The numerical results will be

x

0.5

sin x  

1

0.5

0.5

0.5

3

-0.25

-0.0208333

0.479166667

5

-0.25

0.0002604

0.479427083

7

-0.25

-1.55E-06

0.479425533

9

-0.25

5.382E-09

0.479425539

11

-0.25

-1.223E-11

0.479425539

13

-0.25

1.96E-14

0.479425539

15

-0.25

-2.334E-17

0.479425539

17

-0.25

2.145E-20

0.479425539

19

-0.25

-1.568E-23

0.479425539

You can see that the first 4 terms by themselves give sin .5 correctly to 8 decimal places.