Trying to prove $e$'s irrationality











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Knowing that $limlimits_{xto 0} $$frac{sin(x)}{x}$$= 1$ , $frac{1}{n+1}<n!r_n<frac{1}{n}$, where $r_n=e- sum _{ k=0 }^{ n}{ frac { 1 }{k!}} $



By studying $limlimits_{ntoinfty} nsin(2πn!r_n)$ I have to show that :



$$limlimits_{ntoinfty} nsin(2πn!e)= 2π,$$
and then prove that $e$ is irrational ?










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  • Well, ok. The first limit seems manageable. What are your thoughts?
    – Yuriy S
    Nov 18 at 12:17










  • I tried by composition because $limlimits_{ntoinfty} 2πn!r_n = 0$ and I showed $$limlimits_{ntoinfty} nsin(2πn!r_n)= 2π$$ but i don't think that's correct
    – Lamethyste
    Nov 18 at 12:26












  • Seems fine to me. But not sure about the other limit
    – Yuriy S
    Nov 18 at 12:48










  • Shouldn't the sum in the definition of $r_n$ start at $k=0$ instead of $k=1$?
    – Barry Cipra
    Nov 18 at 12:50










  • You are right, my bad!
    – Lamethyste
    Nov 18 at 12:55















up vote
3
down vote

favorite












Knowing that $limlimits_{xto 0} $$frac{sin(x)}{x}$$= 1$ , $frac{1}{n+1}<n!r_n<frac{1}{n}$, where $r_n=e- sum _{ k=0 }^{ n}{ frac { 1 }{k!}} $



By studying $limlimits_{ntoinfty} nsin(2πn!r_n)$ I have to show that :



$$limlimits_{ntoinfty} nsin(2πn!e)= 2π,$$
and then prove that $e$ is irrational ?










share|cite|improve this question
























  • Well, ok. The first limit seems manageable. What are your thoughts?
    – Yuriy S
    Nov 18 at 12:17










  • I tried by composition because $limlimits_{ntoinfty} 2πn!r_n = 0$ and I showed $$limlimits_{ntoinfty} nsin(2πn!r_n)= 2π$$ but i don't think that's correct
    – Lamethyste
    Nov 18 at 12:26












  • Seems fine to me. But not sure about the other limit
    – Yuriy S
    Nov 18 at 12:48










  • Shouldn't the sum in the definition of $r_n$ start at $k=0$ instead of $k=1$?
    – Barry Cipra
    Nov 18 at 12:50










  • You are right, my bad!
    – Lamethyste
    Nov 18 at 12:55













up vote
3
down vote

favorite









up vote
3
down vote

favorite











Knowing that $limlimits_{xto 0} $$frac{sin(x)}{x}$$= 1$ , $frac{1}{n+1}<n!r_n<frac{1}{n}$, where $r_n=e- sum _{ k=0 }^{ n}{ frac { 1 }{k!}} $



By studying $limlimits_{ntoinfty} nsin(2πn!r_n)$ I have to show that :



$$limlimits_{ntoinfty} nsin(2πn!e)= 2π,$$
and then prove that $e$ is irrational ?










share|cite|improve this question















Knowing that $limlimits_{xto 0} $$frac{sin(x)}{x}$$= 1$ , $frac{1}{n+1}<n!r_n<frac{1}{n}$, where $r_n=e- sum _{ k=0 }^{ n}{ frac { 1 }{k!}} $



By studying $limlimits_{ntoinfty} nsin(2πn!r_n)$ I have to show that :



$$limlimits_{ntoinfty} nsin(2πn!e)= 2π,$$
and then prove that $e$ is irrational ?







limits exponential-function irrational-numbers






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share|cite|improve this question













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share|cite|improve this question








edited Nov 18 at 12:55

























asked Nov 18 at 11:58









Lamethyste

424




424












  • Well, ok. The first limit seems manageable. What are your thoughts?
    – Yuriy S
    Nov 18 at 12:17










  • I tried by composition because $limlimits_{ntoinfty} 2πn!r_n = 0$ and I showed $$limlimits_{ntoinfty} nsin(2πn!r_n)= 2π$$ but i don't think that's correct
    – Lamethyste
    Nov 18 at 12:26












  • Seems fine to me. But not sure about the other limit
    – Yuriy S
    Nov 18 at 12:48










  • Shouldn't the sum in the definition of $r_n$ start at $k=0$ instead of $k=1$?
    – Barry Cipra
    Nov 18 at 12:50










  • You are right, my bad!
    – Lamethyste
    Nov 18 at 12:55


















  • Well, ok. The first limit seems manageable. What are your thoughts?
    – Yuriy S
    Nov 18 at 12:17










  • I tried by composition because $limlimits_{ntoinfty} 2πn!r_n = 0$ and I showed $$limlimits_{ntoinfty} nsin(2πn!r_n)= 2π$$ but i don't think that's correct
    – Lamethyste
    Nov 18 at 12:26












  • Seems fine to me. But not sure about the other limit
    – Yuriy S
    Nov 18 at 12:48










  • Shouldn't the sum in the definition of $r_n$ start at $k=0$ instead of $k=1$?
    – Barry Cipra
    Nov 18 at 12:50










  • You are right, my bad!
    – Lamethyste
    Nov 18 at 12:55
















Well, ok. The first limit seems manageable. What are your thoughts?
– Yuriy S
Nov 18 at 12:17




Well, ok. The first limit seems manageable. What are your thoughts?
– Yuriy S
Nov 18 at 12:17












I tried by composition because $limlimits_{ntoinfty} 2πn!r_n = 0$ and I showed $$limlimits_{ntoinfty} nsin(2πn!r_n)= 2π$$ but i don't think that's correct
– Lamethyste
Nov 18 at 12:26






I tried by composition because $limlimits_{ntoinfty} 2πn!r_n = 0$ and I showed $$limlimits_{ntoinfty} nsin(2πn!r_n)= 2π$$ but i don't think that's correct
– Lamethyste
Nov 18 at 12:26














Seems fine to me. But not sure about the other limit
– Yuriy S
Nov 18 at 12:48




Seems fine to me. But not sure about the other limit
– Yuriy S
Nov 18 at 12:48












Shouldn't the sum in the definition of $r_n$ start at $k=0$ instead of $k=1$?
– Barry Cipra
Nov 18 at 12:50




Shouldn't the sum in the definition of $r_n$ start at $k=0$ instead of $k=1$?
– Barry Cipra
Nov 18 at 12:50












You are right, my bad!
– Lamethyste
Nov 18 at 12:55




You are right, my bad!
– Lamethyste
Nov 18 at 12:55










1 Answer
1






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up vote
1
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accepted










Let $I_n=sumlimits_{k=0}^ndfrac{n!}{k!}in Bbb Z$, then
$$nsin(2pi n! e)=nsin(2pi I_n+2pi n!r_n)=nsin(2pi n!r_n).$$
Note
$$frac 1{n+1}le n!r_n=frac 1{n+1}+frac 1{(n+1)(n+2)}+cdotslesum_{k=1}^inftyfrac1{(n+1)^k}=frac 1n.$$
By sandwich theorem, $n!r_nto 0$ and $ncdot n!r_nto 1$ as $ntoinfty$. Using $sinthetasimtheta$ as $thetato 0$, we get
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}nsin(2pi n! r_n)=2pi.$$





To prove $e$ irrational, assume $e=p/q$, then when $n>q$, $n!einBbb Z$ implies $sin(2pi n! e)equiv 0$, and
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}ncdot 0=0ne 2pi,$$
a contradiction.






share|cite|improve this answer























  • I love you :), you made it clear, thanks!
    – Lamethyste
    Nov 18 at 13:07












  • The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
    – Paramanand Singh
    Nov 18 at 14:07













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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










Let $I_n=sumlimits_{k=0}^ndfrac{n!}{k!}in Bbb Z$, then
$$nsin(2pi n! e)=nsin(2pi I_n+2pi n!r_n)=nsin(2pi n!r_n).$$
Note
$$frac 1{n+1}le n!r_n=frac 1{n+1}+frac 1{(n+1)(n+2)}+cdotslesum_{k=1}^inftyfrac1{(n+1)^k}=frac 1n.$$
By sandwich theorem, $n!r_nto 0$ and $ncdot n!r_nto 1$ as $ntoinfty$. Using $sinthetasimtheta$ as $thetato 0$, we get
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}nsin(2pi n! r_n)=2pi.$$





To prove $e$ irrational, assume $e=p/q$, then when $n>q$, $n!einBbb Z$ implies $sin(2pi n! e)equiv 0$, and
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}ncdot 0=0ne 2pi,$$
a contradiction.






share|cite|improve this answer























  • I love you :), you made it clear, thanks!
    – Lamethyste
    Nov 18 at 13:07












  • The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
    – Paramanand Singh
    Nov 18 at 14:07

















up vote
1
down vote



accepted










Let $I_n=sumlimits_{k=0}^ndfrac{n!}{k!}in Bbb Z$, then
$$nsin(2pi n! e)=nsin(2pi I_n+2pi n!r_n)=nsin(2pi n!r_n).$$
Note
$$frac 1{n+1}le n!r_n=frac 1{n+1}+frac 1{(n+1)(n+2)}+cdotslesum_{k=1}^inftyfrac1{(n+1)^k}=frac 1n.$$
By sandwich theorem, $n!r_nto 0$ and $ncdot n!r_nto 1$ as $ntoinfty$. Using $sinthetasimtheta$ as $thetato 0$, we get
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}nsin(2pi n! r_n)=2pi.$$





To prove $e$ irrational, assume $e=p/q$, then when $n>q$, $n!einBbb Z$ implies $sin(2pi n! e)equiv 0$, and
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}ncdot 0=0ne 2pi,$$
a contradiction.






share|cite|improve this answer























  • I love you :), you made it clear, thanks!
    – Lamethyste
    Nov 18 at 13:07












  • The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
    – Paramanand Singh
    Nov 18 at 14:07















up vote
1
down vote



accepted







up vote
1
down vote



accepted






Let $I_n=sumlimits_{k=0}^ndfrac{n!}{k!}in Bbb Z$, then
$$nsin(2pi n! e)=nsin(2pi I_n+2pi n!r_n)=nsin(2pi n!r_n).$$
Note
$$frac 1{n+1}le n!r_n=frac 1{n+1}+frac 1{(n+1)(n+2)}+cdotslesum_{k=1}^inftyfrac1{(n+1)^k}=frac 1n.$$
By sandwich theorem, $n!r_nto 0$ and $ncdot n!r_nto 1$ as $ntoinfty$. Using $sinthetasimtheta$ as $thetato 0$, we get
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}nsin(2pi n! r_n)=2pi.$$





To prove $e$ irrational, assume $e=p/q$, then when $n>q$, $n!einBbb Z$ implies $sin(2pi n! e)equiv 0$, and
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}ncdot 0=0ne 2pi,$$
a contradiction.






share|cite|improve this answer














Let $I_n=sumlimits_{k=0}^ndfrac{n!}{k!}in Bbb Z$, then
$$nsin(2pi n! e)=nsin(2pi I_n+2pi n!r_n)=nsin(2pi n!r_n).$$
Note
$$frac 1{n+1}le n!r_n=frac 1{n+1}+frac 1{(n+1)(n+2)}+cdotslesum_{k=1}^inftyfrac1{(n+1)^k}=frac 1n.$$
By sandwich theorem, $n!r_nto 0$ and $ncdot n!r_nto 1$ as $ntoinfty$. Using $sinthetasimtheta$ as $thetato 0$, we get
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}nsin(2pi n! r_n)=2pi.$$





To prove $e$ irrational, assume $e=p/q$, then when $n>q$, $n!einBbb Z$ implies $sin(2pi n! e)equiv 0$, and
$$lim_{ntoinfty}nsin(2pi n! e)=lim_{ntoinfty}ncdot 0=0ne 2pi,$$
a contradiction.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 18 at 13:12

























answered Nov 18 at 13:03









Tianlalu

2,8911935




2,8911935












  • I love you :), you made it clear, thanks!
    – Lamethyste
    Nov 18 at 13:07












  • The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
    – Paramanand Singh
    Nov 18 at 14:07




















  • I love you :), you made it clear, thanks!
    – Lamethyste
    Nov 18 at 13:07












  • The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
    – Paramanand Singh
    Nov 18 at 14:07


















I love you :), you made it clear, thanks!
– Lamethyste
Nov 18 at 13:07






I love you :), you made it clear, thanks!
– Lamethyste
Nov 18 at 13:07














The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
– Paramanand Singh
Nov 18 at 14:07






The key idea here is the bound for $r_n$. If $e$ were rational say $m/n$ then $n! e=I_n+n! r_n$ is an integer and this is a contradiction as $n! r_nin[1/(n+1),1/n] $.
– Paramanand Singh
Nov 18 at 14:07




















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