]> Exercise 2.2

## Exercise 2.2

Set up a spreadsheet to compute $exp ⁡ x$ up to say the 100th 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 $x j j !$ in column C (set c2 to =1 and c3 to = c2*b3/a3) and but the "partial sum to the j-th 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 2E-05 2.71827877 9 1 3E-06 2.718281526 10 1 3E-07 2.718281801 11 1 3E-08 2.718281826 12 1 2E-09 2.718281828 13 1 2E-10 2.718281828 14 1 1E-11 2.718281828 15 1 8E-13 2.718281828

Additional Exercise: for $x = 5$ which is the largest term in the series? For $x = 10$ ? In general?