# Taylor's Theorem

Please see the attached file for the fully formatted problems.

1 Write down the Taylor polynomial of , . Use Taylor's theorem to

show that =

2. Given that , 1 < e < 3. Do the followings.

a) Write down the Taylor polynomial of and show that for and

b) Use a) to show that for all c) Conclude that e is irrational?

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#### Solution Preview

1. Proof:

Let , then , , and so on.

In general, we have . By Taylor's Theorem, we expand at and get . Especially, set , we have

.

2. Proof:

(a) Let , then and thus . By Taylor's Theorem, we ...

#### Solution Summary

Taylor's theorem are investigated.

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