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Taylor Expansion and Remainder

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Please see the attached file. Note: I think the formula of h(x) is wrong, so I corrected it and proved by using the new formula.

1. Let
a) Show that =0 for any
NOTE: I think the formula for h(x) should be

Otherwise, the following statements are FALSE.

We consider three cases.

Case 1: k=0

Case 2: k>0

Let y=1/x. Then when x tends to 0, y goes to infinity. Hence,

This is the type of . By the L'Hospital rule, we have

Solution Summary

Taylor Expansion and Remainder are investigated.