Nice proof! Although now I wonder just how different it really is from this one. In Proofs from the Book that technique is to show that $e^2$ and with some adjustments, $e^4$ are irrational.

One can certainly take other sums where the proof you mention applies, although one has to be more careful than I initially thought:

Flawed conjecture (see the comments for a counterexample) You could take any convergent infinite sum $\sum \frac{n_i}{d_i}$ where $\gcd(n_i,d_i)=1$ , and for every integer $q \gt 1$ there is a $j$ so that $q$ divides $d_i$ for all $i \gt j$.

Strengthen the conditions to

• $d_1 | d_2 | \cdots$
• for every integer $q \gt 1$ there is an $i$ so that $q$ divides $d_i$
• $0 \lt |\frac{n_i}{d_i}| \lt \frac{1}{d_{i-1}}$

and the proof does seem to go through. A question is which such sums give interesting real numbers.

$\cos(1)=1-1/2+1/24-1/720\cdots$ works although $\cos(1)=\frac{e^i+e^{-i}}{2}$ so it depends what counts as a trivial relation.

Nice proof! Although now I wonder just how different it really is from this one. In Proofs from the Book that technique is to show that $e^2$ and with some adjustments, $e^4$ are irrational.

One can certainly take other sums where the proof you mention applies, although one has to be more careful than I initially thought:

Flawed conjecture (see the comments for a counterexample) You could take any convergent infinite sum $\sum \frac{n_i}{d_i}$ where $\gcd(n_i,d_i)=1$ , and for every integer $q \gt 1$ there is a $j$ so that $q$ divides $d_i$ for all $i \gt j$.

Strengthen the conditions to

• $d_1 | d_2 | \cdots$
• for every integer $q \gt 1$ there is an $i$ so that $q$ divides $d_i$
• $0 \lt |\frac{n_i}{d_i}| \lt \frac{1}{d_{i-1}}$

and the proof does seem to go through. A question is which such sums give interesting real numbers.

$\cos(1)=1-\frac{1}{2}+\frac{1}{24}-\frac{1}{720}\cdots$ works, although $\cos(1)=\frac{e^i+e^{-i}}{2}$ so it depends what counts as a trivial relation. Also $\cos(\frac{1}{k}).$

Nice proof! You could take any convergent infinite sum $\sum \frac{n_i}{d_i}$ where $\gcd(n_i,d_i)=1$ , and for every integer $q \gt 1$ there is a $j$ so that $q$ divides $d_i$ for all $i \gt j$. For example $d_i$ is the least common multiple of the integers up to $i$ and $n_i$ is the first prime greater than $i$.The question is if there are nice reals with such a series.

I suppose that $\cos(1)=1-1/2+1/24-1/720\cdots$ works although $\cos(1)=\frac{e^i+e^{-i}}{2}$ so it depends what counts as a trivial relation. Also, $\sec(1)$ looks ok.

Nice proof! Although now I wonder just how different it really is from this one. In Proofs from the Book that technique is to show that $e^2$ and with some adjustments, $e^4$ are irrational.

One can certainly take other sums where the proof you mention applies, although one has to be more careful than I initially thought:

Flawed conjecture (see the comments for a counterexample) You could take any convergent infinite sum $\sum \frac{n_i}{d_i}$ where $\gcd(n_i,d_i)=1$ , and for every integer $q \gt 1$ there is a $j$ so that $q$ divides $d_i$ for all $i \gt j$.

Strengthen the conditions to

• $d_1 | d_2 | \cdots$
• for every integer $q \gt 1$ there is an $i$ so that $q$ divides $d_i$
• $0 \lt |\frac{n_i}{d_i}| \lt \frac{1}{d_{i-1}}$

and the proof does seem to go through. A question is which such sums give interesting real numbers.

$\cos(1)=1-1/2+1/24-1/720\cdots$ works although $\cos(1)=\frac{e^i+e^{-i}}{2}$ so it depends what counts as a trivial relation.