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**acevipa** 1) If $\displaystyle e$ were rational, it would be of the form $\displaystyle e=\frac{p}{q}$, where $\displaystyle p$ and $\displaystyle q$ are positive integers. Select an integer $\displaystyle k$ such that $\displaystyle k\geq 3$ and $\displaystyle k\geq q$. Use Taylor's theorem to show that

$\displaystyle \frac{p}{q}=e=1+\frac{1}{1!}+\frac{1}{2!}+...+\fra c{1}{k!}+\frac{e^z}{(k+1)!}$

for some $\displaystyle z\in (0,1)$

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

Yes.

2) Suppose that:

$\displaystyle s_k=1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{k!}$

Show that $\displaystyle k!(e-s_k)$ is an integer

Check separatedly $\displaystyle \,\,k!e\,,\,k!s_k$

3) Show that $\displaystyle 0<k!(e-s_k)<1$

For this I think you need to already know that $\displaystyle e<4$ and do some simple algebra.

4) Conclude that $\displaystyle e$ is irrational

Obvious from the above.

Tonio