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Unformatted text preview: orem directly to e−x . The simple answer to this question is
DE Math 128 330 (B. Forrest)2 CHAPTER 4. Sequences and Series 4.5. Introduction to Taylor Series that the method we outlined above results in much easier calculations. Because all of the
derivatives of eu are just eu , it was rather easy to obtain an upper bound for the error in
using linear approximation to estimate eu . This then gave us a very quick estimate of the
diﬀerence between 1 − x2 and e−x . If we wanted to apply Taylor’s Theorem directly to
e−x , viewing 1 − x2 as the third degree Taylor polynomial for more accuracy, we would
need to be able to determine how large the fourth derivative of e−x might be. This is not
an easy task since the fourth derivative of e−x is the function 12e−x − 48x2 e−x +16x4 e−x .
The method we outlined avoids this diﬃculty.
We will now see how Taylor’s Theorem can help in calculating various limits. In order to
simplify the notation, we will only consider li...
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