Looking for a proof that $pi$ is irrational using a series representation.
$begingroup$
I'm searching for a proof that $pi$ is irrational using a series representation for $pi$. I've seen that Apery proved that $zeta(2)$ is irrational by using the series
begin{align}
zeta(2) = frac{pi^2}{6} = 3 sum_{n=1}^{infty} frac{1}{n^2binom{2n}{n}}
end{align}
but can't find the proof. Since proving that $pi^2$ is irrational also shows that $pi$ is irrational this proof is interesting.
This is the only proof I've heard of (but can't find it) that uses a series for $pi$.
number-theory irrational-numbers pi
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
$endgroup$
add a comment |
$begingroup$
I'm searching for a proof that $pi$ is irrational using a series representation for $pi$. I've seen that Apery proved that $zeta(2)$ is irrational by using the series
begin{align}
zeta(2) = frac{pi^2}{6} = 3 sum_{n=1}^{infty} frac{1}{n^2binom{2n}{n}}
end{align}
but can't find the proof. Since proving that $pi^2$ is irrational also shows that $pi$ is irrational this proof is interesting.
This is the only proof I've heard of (but can't find it) that uses a series for $pi$.
number-theory irrational-numbers pi
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
$endgroup$
$begingroup$
What about these?
$endgroup$
– Rhys Hughes
Dec 31 '18 at 19:29
$begingroup$
But is there any proof that $pi$ is irrational using those series?
$endgroup$
– Pinteco
Dec 31 '18 at 19:55
add a comment |
$begingroup$
I'm searching for a proof that $pi$ is irrational using a series representation for $pi$. I've seen that Apery proved that $zeta(2)$ is irrational by using the series
begin{align}
zeta(2) = frac{pi^2}{6} = 3 sum_{n=1}^{infty} frac{1}{n^2binom{2n}{n}}
end{align}
but can't find the proof. Since proving that $pi^2$ is irrational also shows that $pi$ is irrational this proof is interesting.
This is the only proof I've heard of (but can't find it) that uses a series for $pi$.
number-theory irrational-numbers pi
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
$endgroup$
I'm searching for a proof that $pi$ is irrational using a series representation for $pi$. I've seen that Apery proved that $zeta(2)$ is irrational by using the series
begin{align}
zeta(2) = frac{pi^2}{6} = 3 sum_{n=1}^{infty} frac{1}{n^2binom{2n}{n}}
end{align}
but can't find the proof. Since proving that $pi^2$ is irrational also shows that $pi$ is irrational this proof is interesting.
This is the only proof I've heard of (but can't find it) that uses a series for $pi$.
number-theory irrational-numbers pi
number-theory irrational-numbers pi
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
asked Dec 31 '18 at 19:21
PintecoPinteco
731313
731313
$begingroup$
What about these?
$endgroup$
– Rhys Hughes
Dec 31 '18 at 19:29
$begingroup$
But is there any proof that $pi$ is irrational using those series?
$endgroup$
– Pinteco
Dec 31 '18 at 19:55
add a comment |
$begingroup$
What about these?
$endgroup$
– Rhys Hughes
Dec 31 '18 at 19:29
$begingroup$
But is there any proof that $pi$ is irrational using those series?
$endgroup$
– Pinteco
Dec 31 '18 at 19:55
$begingroup$
What about these?
$endgroup$
– Rhys Hughes
Dec 31 '18 at 19:29
$begingroup$
What about these?
$endgroup$
– Rhys Hughes
Dec 31 '18 at 19:29
$begingroup$
But is there any proof that $pi$ is irrational using those series?
$endgroup$
– Pinteco
Dec 31 '18 at 19:55
$begingroup$
But is there any proof that $pi$ is irrational using those series?
$endgroup$
– Pinteco
Dec 31 '18 at 19:55
add a comment |
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$begingroup$
What about these?
$endgroup$
– Rhys Hughes
Dec 31 '18 at 19:29
$begingroup$
But is there any proof that $pi$ is irrational using those series?
$endgroup$
– Pinteco
Dec 31 '18 at 19:55