imported>Voidxor: No need to subst the {{Userbox}} template (doing so makes life difficult). Add sentence period.

2018-02-03T00:55:09Z

No need to subst the {{Userbox}} template (doing so makes life difficult). Add sentence period.

New page

{{Userbox
| id = <math>e \ne \frac {p} {q}</math>
| id-s = 16
| id-c = #ffffff
| info = This user can prove that the number, <big>'''''e'''''</big>, is irrational.
| info-a = center
| info-c = #F3F3F3
| info-fc = #505080
| info-s = 10
| border-c = black
}}

<noinclude>
{{Clear}}

'''Usage: <nowiki>{{User:Karlhahn/User e-irrational}}</nowiki>'''

'''PROOF:'''

If <math>\scriptstyle e</math> were rational, then

:<math>e = \frac {p} {q}</math>

where <math>\scriptstyle p</math> and <math>\scriptstyle q</math> are both positive integers. Hence

:<math>q e = p</math>

making <math>\scriptstyle qe</math> an integer. Multiplying both sides by <math>\scriptstyle (q-1)!</math>,

:<math>q! e = p(q-1)!</math>

so clearly <math>\scriptstyle q! e</math> is also an integer. By [[Maclaurin series]]

:<math>e = \sum_{k=0}^\infty \frac {1} {k!}</math>

Multiplying both sides by <math>\scriptstyle q!</math>:

:<math>q! e = \sum_{k=0}^\infty \frac {q!} {k!}</math>

The first <math>\scriptstyle q+1</math> 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

:<math>r = \sum_{k=1}^\infty \frac {q!} {(q+k)!}</math>
making <math>\scriptstyle r</math> an integer. Observe that

:<math>\frac {q!} {(q+k)!} < \frac {q!} {q! q^k} = \frac {1} {q^k}</math>

So

:<math>r = \sum_{k=1}^\infty \frac {q!} {(q+k)!} < \sum_{k=1}^\infty \frac {1} {q^k}</math>

But

:<math>\sum_{k=1}^\infty \frac {1} {q^k} = \frac {1} {q-1}</math>

This means that <math>\scriptstyle 0 < r < \frac {1} {q-1} \le 1</math>, which requires that <math>\scriptstyle r</math> be an integer between zero and one. That is clearly impossible, hence <math>\scriptstyle e</math> is irrational
[[Category:Mathematics user templates|Irrational e]]
</noinclude>

User:Karlhahn/User e-irrational - Revision history

imported>Voidxor: No need to subst the {{Userbox}} template (doing so makes life difficult). Add sentence period.