1) If were rational, it would be of the form , where and are positive integers. Select an integer such that and . Use Taylor's theorem to show that

for some

Is this just the Taylor series about 0. and the end part is the Lagrange formula for the remainder?

Yes.

2) Suppose that:

Show that is an integer

Check separatedly

3) Show that

For this I think you need to already know that and do some simple algebra.

4) Conclude that is irrational

Obvious from the above.

Tonio

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