User:Karlhahn/User e-irrational

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This user can prove that the number, e, is irrational.

Usage: {{User:Karlhahn/User e-irrational}}


If were rational, then

where and are both positive integers. Hence

making an integer. Multiplying both sides by ,

so clearly is also an integer. By Maclaurin series

Multiplying both sides by :

The first terms of this sum are integers. It follows that the sum of the remaining terms must also be an integer. The sum of those remaining terms is

making an integer. Observe that



This means that , which requires that be an integer between zero and one. That is clearly impossible, hence is irrational