Home  18.013A  Chapter 2  Section 2.3 


Set up a spreadsheet to compute the $\mathrm{sin}x$ for any inputted $x$ . How many terms in the $\mathrm{sin}x$ power series expansion do you need to evaluate $\mathrm{sin}.5$ to 8 decimal places?
Solution:
The power series expansion of $\mathrm{sin}x$ consists of the odd power terms in the expansion of $\mathrm{exp}x$ , 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 $\mathrm{exp}x$ 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 
$\mathrm{sin}x$  
1 
=B1 
=A2*B2 
=D1+C2 
=A2+2 
=B2*B2 
=C2*B3/A3/(A31) 
=D2+C3 
=A3+2 
=B3 
=C3*B4/A4/(A41) 
=D3+C4 
=A4+2 
=B4 
=C4*B5/A5/(A51) 
=D4+C5 
=A5+2 
=B5 
=C5*B6/A6/(A61) 
=D5+C6 
=A6+2 
=B6 
=C6*B7/A7/(A71) 
=D6+C7 
=A7+2 
=B7 
=C7*B8/A8/(A81) 
=D7+C8 
=A8+2 
=B8 
=C8*B9/A9/(A91) 
=D8+C9 
=A9+2 
=B9 
=C9*B10/A10/(A101) 
=D9+C10 
=A10+2 
=B10 
=C10*B11/A11/(A111) 
=D10+C11 
=A11+2 
=B11 
=C11*B12/A12/(A121) 
=D11+C12 
=A12+2 
=B12 
=C12*B13/A13/(A131) 
=D12+C13 
=A13+2 
=B13 
=C13*B14/A14/(A141) 
=D13+C14 
=A14+2 
=B14 
=C14*B15/A15/(A151) 
=D14+C15 
=A15+2 
=B15 
=C15*B16/A16/(A161) 
=D15+C16 
=A16+2 
=B16 
=C16*B17/A17/(A171) 
=D16+C17 
=A17+2 
=B17 
=C17*B18/A18/(A181) 
=D17+C18 
=A18+2 
=B18 
=C18*B19/A19/(A191) 
=D18+C19 
=A19+2 
=B19 
=C19*B20/A20/(A201) 
=D19+C20 
=A20+2 
=B20 
=C20*B21/A21/(A211) 
=D20+C21 
=A21+2 
=B21 
=C21*B22/A22/(A221) 
=D21+C22 
The numerical results will be
$x$ 
0.5 
$\mathrm{sin}x$  
1 
0.5 
0.5 
0.5 
3 
0.25 
0.0208333 
0.479166667 
5 
0.25 
0.0002604 
0.479427083 
7 
0.25 
1.55E06 
0.479425533 
9 
0.25 
5.382E09 
0.479425539 
11 
0.25 
1.223E11 
0.479425539 
13 
0.25 
1.96E14 
0.479425539 
15 
0.25 
2.334E17 
0.479425539 
17 
0.25 
2.145E20 
0.479425539 
19 
0.25 
1.568E23 
0.479425539 
You can see that the first 4 terms by themselves give $\mathrm{sin}.5$ correctly to 8 decimal places.
