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Home Infinite Series Taylor's Formula Examples Example 2: MacLaurin's Formula
 
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Example 2: MacLaurin's Formula

Find MacLaurin's Formula for f(x) = ex.

The nth derivative is f(n)(x) = ex f(n)(0) = 1. MacLaurin's Formula is

09_infinite_series-694.gif09_infinite_series-695.gif

for some tn between 0 and x. For t between 0 and x the value of et is always less than or equal to 3|x|, for et ≤ e|x| ≤ 3|x|.

We therefore have the formula

(3)

09_infinite_series-696.gif, 09_infinite_series-697.gif

The formula (3) can be used to approximate ex. Let us set x = 1 and approximate e. The error estimate is now

09_infinite_series-698.gif

n

1/n!

Approximate value for e

09_infinite_series-699.gif

Error estimate

09_infinite_series-700.gif

2

0.500000

2.500000

0.500000

3

0.166667

2.666667

0.125000

4

0.041667

2.708333

0.025000

5

0.008333

2.716667

0.004167

6

0.001389

2.718056

0.000594

7

0.000198

2.718254

0.000075

8

0.000025

This compares with e = 2.718282.
Home Infinite Series Taylor's Formula Examples Example 2: MacLaurin's Formula

Last Update: 2006-11-15