Home  18.013A  Chapter 2  Section 2.1 


Set up a spreadsheet to compute $\mathrm{exp}x$ up to say the 100^{th} term using your series.
Solution:
What I would do is to put the value of $x$ (which you can specify as you wish) in b1. (and the symbol $x$ in a1)
Then I would put $j$ in column A starting with 0 in a2. (Put =a2+1 in a3 and copy it down) ; put $x$ in column B (set =b1+1 in b2 and copy it down); put $\frac{{x}^{j}}{j!}$ in column C (set c2 to =1 and c3 to = c2*b3/a3) and but the "partial sum to the jth term" in column D. (set d2=d1+c2 and copy it down)
Setting $x=1$ here is what I get
x 
1 

0 
=B1 
1 
=D1+C2 
=A2+1 
=B2 
=C2*B3/A3 
=D2+C3 
=A3+1 
=B3 
=C3*B4/A4 
=D3+C4 
=A4+1 
=B4 
=C4*B5/A5 
=D4+C5 
=A5+1 
=B5 
=C5*B6/A6 
=D5+C6 
=A6+1 
=B6 
=C6*B7/A7 
=D6+C7 
=A7+1 
=B7 
=C7*B8/A8 
=D7+C8 
=A8+1 
=B8 
=C8*B9/A9 
=D8+C9 
=A9+1 
=B9 
=C9*B10/A10 
=D9+C10 
=A10+1 
=B10 
=C10*B11/A11 
=D10+C11 
=A11+1 
=B11 
=C11*B12/A12 
=D11+C12 
=A12+1 
=B12 
=C12*B13/A13 
=D12+C13 
=A13+1 
=B13 
=C13*B14/A14 
=D13+C14 
The results are
$x$ 
1 

0 
1 
1 
1 
1 
1 
1 
2 
2 
1 
0.5 
2.5 
3 
1 
0.1667 
2.666666667 
4 
1 
0.0417 
2.708333333 
5 
1 
0.0083 
2.716666667 
6 
1 
0.0014 
2.718055556 
7 
1 
0.0002 
2.718253968 
8 
1 
2E05 
2.71827877 
9 
1 
3E06 
2.718281526 
10 
1 
3E07 
2.718281801 
11 
1 
3E08 
2.718281826 
12 
1 
2E09 
2.718281828 
13 
1 
2E10 
2.718281828 
14 
1 
1E11 
2.718281828 
15 
1 
8E13 
2.718281828 
Additional Exercise: for $x=5$ which is the largest term in the series? For $x=10$ ? In general?
