Edit made on May 09, 2019 by ColinWright at 22:23:22
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HEADERS_END
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
$e=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots$
So, suppose that EQN:e=m/n.
So, suppose that $e=m/n.$
Multiply both sides by EQN:n! .
Multiply both sides by $n!$.
The left-hand side becomes an integer.
The first EQN:n+1 terms of the right-hand side become integers.
The first $n+1$ 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 and therefore not an integer.
The rest of the right-hand side is $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots$
which is positive but smaller than 1 and therefore not an integer.
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
Prove that $\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+\frac{1}{(n+1)(n+2)(n+3)}+\cdots<1$