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Home | 18.013A | Chapter 2 | Section 2.3 |
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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,
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 -x2 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 |
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3 |
-0.25 |
-0.0208333 |
0.479166667 |
|
5 |
-0.25 |
0.0002604 |
0.479427083 |
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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 |
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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.
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