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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/ :

EQN:e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots

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

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

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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