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
Is this just the Taylor series about 0. and the end part is the Lagrange formula for the remainder?
2) Suppose that:
Show that is an integer
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.