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    Taylor's Theorem

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    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.