| Version current |
Version 4 |
| We have the series |
We have the series |
| \[ |
\[ |
| e^{-1} = \sum_{k=0}^\infty {(-1)^k \over k!} |
e^{-1} = \sum_{k=0}^\infty {(-1)^k \over k!} |
| \] |
\] |
| Note that this is an alternating series and that the magnitudes of the |
Note that this is an alternating series and that the magnitudes of the |
| terms decrease. Hence, for every integer $n > 0$, we have the bound |
terms decrease. Hence, for every integer $n > 0$, we have the bound |
| \[ |
\[ |
| 0 < |
0 < |
| \left| \sum_{k=0}^{n} {(-1)^k \over k!} - e^{-1} \right| < |
\left| \sum_{k=0}^{n} {(-1)^k \over k!} - e^{-1} \right| < |
|
{1 \over (n+1)!},
|
{1 \over (n+1)!} |
| \] |
\] |
| by the \PMlinkname{Leibniz' estimate for alternating |
Assume that $e = n/m$, where $m$ and $n$ are integers and $n > 0$. |
| series}{LeibnizEstimateForAlternatingSeries}.\, Assume |
|
| that $e = n/m$, where $m$ and $n$ are integers and $n > 0$.\, |
|
| Then we would have |
Then we would have |
| \[ |
\[ |
| 0 < |
0 < |
| \left| \sum_{k=0}^{n} {(-1)^k \over k!} - |
\left| \sum_{k=0}^{n} {(-1)^k \over k!} - |
| {m \over n} \right| < |
{m \over n} \right| < |
| {1 \over (n+1)!} . |
{1 \over (n+1)!} . |
| \] |
\] |
| Multiplying both sides by $n!$, this would imply |
Multiplying both sides by $n!$, this would imply |
| \[ |
\[ |
| 0 < |
0 < |
| \left| \sum_{k=0}^{n} {(-1)^k n! \over k!} - |
\left| \sum_{k=0}^{n} {(-1)^k n! \over k!} - |
| m (n-1)! \right| < |
m (n-1)! \right| < |
| {1 \over n+1} , |
{1 \over n+1} , |
| \] |
\] |
| which is a contradiction because every term in the sum is an integer, |
which is a contradiction because every term in the sum is an integer, |
| but there are no integers between $0$ and $1/(n+1)$. |
but there are no integers between $0$ and $1/(n+1)$. |
