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) |
Home | 18.013A | Chapter 2 | Section 2.1 |
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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
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
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x |
1 |
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0 |
=B1 |
1 |
=D1+C2 |
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=A2+1 |
=B2 |
=C2*B3/A3 |
=D2+C3 |
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=A3+1 |
=B3 |
=C3*B4/A4 |
=D3+C4 |
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=A4+1 |
=B4 |
=C4*B5/A5 |
=D4+C5 |
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=A5+1 |
=B5 |
=C5*B6/A6 |
=D5+C6 |
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=A6+1 |
=B6 |
=C6*B7/A7 |
=D6+C7 |
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=A7+1 |
=B7 |
=C7*B8/A8 |
=D7+C8 |
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=A8+1 |
=B8 |
=C8*B9/A9 |
=D8+C9 |
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=A9+1 |
=B9 |
=C9*B10/A10 |
=D9+C10 |
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=A10+1 |
=B10 |
=C10*B11/A11 |
=D10+C11 |
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=A11+1 |
=B11 |
=C11*B12/A12 |
=D11+C12 |
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=A12+1 |
=B12 |
=C12*B13/A13 |
=D12+C13 |
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=A13+1 |
=B13 |
=C13*B14/A14 |
=D13+C14 |
The results are
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x |
1 |
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0 |
1 |
1 |
1 |
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1 |
1 |
1 |
2 |
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2 |
1 |
0.5 |
2.5 |
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3 |
1 |
0.1667 |
2.666666667 |
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4 |
1 |
0.0417 |
2.708333333 |
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5 |
1 |
0.0083 |
2.716666667 |
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6 |
1 |
0.0014 |
2.718055556 |
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7 |
1 |
0.0002 |
2.718253968 |
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8 |
1 |
2E-05 |
2.71827877 |
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9 |
1 |
3E-06 |
2.718281526 |
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10 |
1 |
3E-07 |
2.718281801 |
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11 |
1 |
3E-08 |
2.718281826 |
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12 |
1 |
2E-09 |
2.718281828 |
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13 |
1 |
2E-10 |
2.718281828 |
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14 |
1 |
1E-11 |
2.718281828 |
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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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