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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?

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