Most recent change of EIsIrrational

Edit made on February 13, 2009 by DerekCouzens at 07:44:41

Deleted text in red / Inserted text in green

It is much easier to prove that /e/ (the base of natural logarithms) is irrational than that pi is irrational. The key is the famous series for /e/ :


So, suppose that EQN:e=m/n.

Multiply both sides by EQN:n! .

The left-hand side becomes an integer.

The first EQN:n+1 terms of the right-hand side become integers.

The rest of the right-hand side is EQN:\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots which is positive but smaller than 1 (EXERCISE: prove this!) and therefore not an integer.

So, integer = integer + not-integer; contradiction.


Enrichment task

Prove that EQN:\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots<1